I'm going to lay out an easy, step-by-step method that is in parts inspired by and borrowed from a book of plots I own. What you'll get as a result is a situation to drop your players into and see what happens.
Eureka by Engine Publishing is an excellent book of adventure plots that are designed to be adaptable to any game and any genre. The format of the book's plots requires a method for creating specific adventures from general plots, which it provides specific guidance for. It also explains the structure of its plots, explaining how they're made so that they're so easily adaptable.
Adapting Beowulf breaks down into two large steps that are perfect analogues from the book:
- Strip the story down to its essential components and themes.
- Build onto the stripped-down plot to make it a specific RPG adventure.
Or, to put it simply: generalise the story, then re-specify the story.
Each of those two steps obviously have sub-steps, which I'll cover below, borrowing from the methodology presented in Eureka.
1. Strip out the details
The plots are presented in a generalised form: NPCs aren't names but plot-related roles such as "the mayor hiding the town's secret" or "the assassin ritually killing city folk to their dark god"; kinds of places that are defined functionally, such as "a ruined town" or "a space station run by an low-level machine intelligence"; and so on. They also have themes, such as "Crimes of Love", "Disaster", and "Self-Sacrifice for an Ideal".
1a. Identify the themes of the story you want to emphasise
Beowulf has many possible themes, being such an old story retold countless times. Pick the theme that you find interesting and that you think will be useful as a guide for your later decisions in this process. Don't hesitate to revisit this step if necessary as your work progresses if another theme comes to the fore or would work better.
1b. Identify character roles
Beowulf (in one of its variations) has the foreign hero, the king with a secret, the monster-son of the king, the witch-mother of the monster, and a general pile of warriors and non-warrior townspeople.
If you're going to adapt a story across genres you can further generalise those roles either now, or easily enough in your head during stage 2. For the sake of example though, this could be further generalised as: the New Arrival Who Threatens the Status Quo, the Authority With an Authority-Threatening Secret, the One Wronged by The Authority's Secret, the Pariah/Exile Who Knows the Secret, and the People Who Depend On and Support the Authority. Those roles could easily be adapted to sci-fi, horror, post-apocalypse, whatever.
1c. Identify places
Functional places in Beowulf are the small kingdom, the monster's lair, the witch's lair.
1d. Identify plot turns
The monster threatens the kingdom. The hero arrives. The king tries to hide the secret. The monster attacks, revealing that the kingdom is in trouble. The witch attempts to use the hero for her own agenda that is contrary to the kingdom's welfare. The hero has the opportunity to confront the monster in a final way—which could be accepted, or not.
- Also in the step you should identify a hook if it's obvious. However, this will often depend on the specifics of your PCs and the adventure you later build, so skipping this is OK. You'll have the chance to figure out the hook again when building the plot into an adventure.
It's worth noting that the output of this step doesn't have a corresponding step in the building stage of the process. These plot threads aren't your adventure! Like everything in stage (1), the point is to get a clear idea of how the source story works. However, even if these plot turns aren't formally used, having clearly looked at them, they will inevitably feed into the process of inspiration and creation in stage (2).
The point of all this isn't to generate a rigid plot to march your players through "Beowulf in space" anyway. No, the point is to generate a rich situation for your players to interact with in dynamic ways, and give you enough material so that can deal with anything that might come up during play, despite not having a fixed plot to follow.
2. Build the plot into an adventure
This is taken with significant inspiration from the text of Eureka. I can't recommend this book enough for GMs who are looking for tools to adapt arbitrary plots to their game, and the rest of the 500+ plots are excellent material with much of the work already done. Do go check out the book. :)
Importantly, during the building process you might find that you don't need a particular piece from your stripped-down story. Don't worry about it! If you've got enough to be an interesting adventure, leave out bits that aren't useful.
2a. Cast your NPCs
You've got a foreign hero, king, witch, monster, and townspeople. Find characters that are already in your campaign that could fill these roles. Perhaps your PCs are near a town with an interesting NPC mayor you've already developed; use her as the "king" role, which will naturally start you adapting the rest of the plot to suit. You've already got townspeople then.
Ask yourself about the mayor's past: is there an obvious secret? If not, give them one. Is there an obvious person who would embody the secret who could be an existential threat to the town? Probably, but if not it should be easy at this point to think of one. Detail it, working out combat stats if necessary. (You'll probably already be thinking of locations for these NPCs, which is fine—you don't have to wait for the later steps.)
Finally, you'll want to figure out who is the exile who knows the mayor's secret. Perhaps the hermit brother of the mayor, exiled on pain of death by the mayor for threatening to reveal the secret years ago. Did the mayor do it for petty, personal, unscrupulous reasons? Or did she do it because the secret was so dire that the town must be saved from it and the brother was the one being unethical? Or perhaps they both were being ethical, but with conflicting priorities? Or are they both just nasty rat-bastards?
These kind of details of motivation will make your adventure easy to run at the table, so they're worth thinking about as inspiration strikes you.
When considering these questions (and ever question in this process, really) it is helpful to keep the theme you're working with in mind—often it will give you the answer straight off, and when it doesn't it gives you a direction to think in, narrowing down your creative possibilities and making it easier to answer the question you post yourself.
2b. Fix your locations
Every adventure needs interesting locations to adventure in. You've got a "monster" lair to detail, as well as the surrounding area where it's hidden. Is it inside the town? Under it? In the surrounding land? If the "monster" who holds the mayor's secret is more mundane, perhaps they are a simple townsperson, and you just need to figure out where they live so you've got a backdrop when the PCs seek them out.
For the sake of argument, let's say that the secret is the mayor has an eldest son from a star-crossed elopement from when she was young, her husband was dragged off by his family back to their homeland, she had the child in secret, and the brother knows this. But to become mayor according to the town's religion, she had to be unmarried so she could symbolically "marry" the town's patron god, and at the time she was the only person that could unite the warring guilds of the town. So she maintains the peace of the town, her son is ignorant of his parentage but is stirring up the guilds into conflict again, and the mayor's exiled brother believes that uncovering the heresy that the mayor is living is the path to saving the town, while the mayor believes that only she can keep the guilds at peace and the heresy is a necessary evil.
Whew, that was a digression. But it does illustrate the strength of this method: Just by asking yourself about where to put the monster's lair (or whatever), suddenly puzzle pieces start fitting together and you've got an entire plot, already in motion, just waiting for the PCs to step in and kick the plot into action.
2c. Develop your hook(s)
You've got a NPCs and locations, you've probably already got large pieces of plot, if not an entire plot already. Now find some entry point that suits your existing group of PCs: a hook, and the opening you'll use to introduce that hook.
If you are using existing NPCs in your plot roles, you can use existing relationships the PCs have—in fact, this is the best argument for using existing NPCs to build your plot into an adventure. Perhaps they know the mayor, and while passing through they are approached by a runner who gives them an invitation to dinner. Even without relationships there are plenty of juicy hooks to be had: the PCs witness guild members clashing in the street, and notice they have a charismatic leader. Or they camp outside of town, intending to pass by, and the hermit brother approaches their campfire with news of strife and a request for their aid in addressing a grave spiritual problem threatening the town.
You'll notice that in all these cases, the PCs are fulfilling the role of the foreign hero of the stripped-down Beowulf. This need not be the case. If you're playing a kings and knights sort of game, your PCs might be the authority figure / king / mayor, and the secret is actually between a visiting foreign hero and the witch and monster. Alternatively you could have the foreign hero role be an NPC as well, and have the PCs come into the situation while it's already in motion.
(In fact, you could create an interesting pregenerated character one-shot game by creating all the roles yourself, creating PCs to fit them, and then letting your players play out the roles of the hero, king, witch, and monster. There are several systems suited to such play†, and even an entire game, Love in the Time of Seið, build around that kind of plot.)
Whatever plot role you've left open for the players' characters, you should be able to come up with a variety of hooks and some ideas of how to reveal them, giving you the flexibility to expose them to the adventure in a natural way without railroading them (unless they like that, of course).
By now you will have successfully build an adventure inspired by the stripped-down plot of the original story. Throughout the above steps, you might have stuck close to the original plot, or you might have deviated significantly to the point where it's barely recognisable even knowing what it's based on. Either way, you have a lovely adventure to run now!
Depending on your GMing style and the state of your campaign, you may want to develop a few more items.
You may want to develop a few signs that you can use as foreshadowing, if you're not going to run the adventure right away. Perhaps the PCs are passing through a town during another adventure and you know they'll be back this way later: you could foreshadow the guild conflict by having some of the shopkeepers grumble about the other guilds while they're buying supplies, for example.
You chosen gaming system might need specific encounters and stats fleshed out beforehand. You've got a clear large picture, so it's just a matter of picking a few conflicts for the PCs to get caught up in or tackle. Perhaps you have a skill-based conflict where they try to save some bystanders during a guild conflict that turns bloody or arson-y. Maybe someone sense thugs after the PCs at night to "persuade" them that they should keep their noses out of the town's business—this could be a guild leader, the secret son of the mayor, the mayor, or even the hermit, all depending on what motives they have and who you want to highlight as the opposition to the PCs. And of course there's always room for a climactic encounter with the monster-role NPC, which could be a roiling battle, a private duel, or even a high-stakes argument.
Though the process might seem involved, it's really very simple at it's core. Identify the important moving parts of the story so that you can see them clearly and in isolation, bare of any obscuring details. Then take that plot scaffolding and hang your own favourite places, NPCs, and MacGuffins on it, and enjoy how inspiration for the in the motives and details comes easy and natural while you think of how to hang flesh on the bones.
† I've played such games—where all the players are in different plot roles and aren't necessarily working together—with Burning Wheel and Apocalypse World. Others would work; pretty much any game that rewards players for pursuing their character's motivations instead of working together, even when that gets the character in trouble, would work well for it.