There are several such games, some of which are short-form, but some long-form (i.e., useful for open-ended campaigns) as well. I can think of two of each off the top of my head / that are sitting on my shelf.
Unsurprisingly, all of these are heavy on the story-games style of mechanics, since without randomised resolution and stats, the moving parts of the system stick to mediating the narrative powers of the players instead of directly managing the fictional powers of characters. As a nice bonus, all of them are zero-prep games (apart from reading the rules).
Two long form games
Microscope has already been mentioned. I'll second it as an excellent game that manages to be inexplicably compelling for long-standing gamers and absolute novices alike. It avoids relying on numerical representation and random factors by leaving characters' capabilities up to the group's implicit understanding of the characters; it then manages differences in people's understanding in the same way it handles "what happens next" plot disagreements: a propose-and-vote system. This way everyone gets a say, and the most popular proposal for the next turn of events (and by implication, the capabilities of the characters involved) is the one that happens. (Microscope works smoothes for groups of 3-5 or smaller, due to this mechanic.)
The hitch with Microscope is that it's designed for playing out the vignettes of the turning points in a long history (several centuries up to hundreds of millenia), so characters don't tend to persist beyond one scene. That said, by group agreement you can easily just keep the camera on one group of characters through multiple rounds, and the system doesn't care.
Archipelago is another game that leverages social dynamics and group consensus in its mechanics to make decisions without stats or randomisers. (Also, it's free and short, running only 22 pages of PDF.)
Archipelago has a more traditional one-character, one-player structure, but it's completely different from there. There's no GM, and by default when it's your turn you have the power to control everything about what your character does, can do, and the opposition. However, some GM power is devolved to the other players when it's your turn: they can take specific, system-defined actions that you are required to respond to. You still have control of what happens, but once another player has invoked one of these "GM powers" you have to figure out how to use it. The things they can do include adding opposition, requiring a re-do, making you slow down or not skip over a detail, requiring a follow-up scene, and taking control of an NPC.
For all but taking control of an NPC, the other players' involvement in your turn is limited to just invoking the rule (through a specific phrase) and then backing off. After their interruption, you take back control and incorporate what they required. For example, you might be describing your character battling a trio of orcs you discovered camping in the King's Forest:
You: The orcs are surprised, too focused on their the roasting deer and slowed by the former contents of a keg of beer—undoubtedly stolen—that stands nearby. They're pose no real threat and soon the last one is bleeding out around their guttering campfire.
Player 2: Woah, that was a massacre. Hold up: "That might not be quite so easy." (the player is expecting the orcs wouldn't be defeated that easily)
You: Okay, hm. Actually, I still think they'd be massacred, so… "As the last one bleeds out, a crash of shattering branches barely announces the troll that suddenly thunders into the small clearing, swinging a massive oak club as it comes. The orcs weren't alone! She throws herself out of the way, knowing her arm would likely shatter if she was foolish enough to try blocking…"
Notice that, despite how the interrupting player has an expectation of what "harder" means, the decision of how and what is "not so easy" stays in the hands of the current player.
The mechanics are set up in such a way that you always eventually have final say in how your character performs, but if the action you're describing isn't winning over your fellow players they get to ask you to try something more interesting. The net effect is that the story ends up refined and improved on-the-fly, resulting in something that is better than what any one person would have come up with on their own, but each person still has ultimate authority over what their character ends up doing.
The other phrases are:
"Try a different way" — This is used whenever the current player's turn involves something you don't want to happen, or was narrated in a way that you don't think fits. It can be used to enforce genre conventions (e.g. "Try a different way: can we not have slapstick goblins in our gritty fantasy?"), or to veto things the player has your character do on their turn ("Try a different way. I like how you made my character bad-ass there, but I don't want them to reveal themselves to the City Watch so blatantly.")
"Describe that in detail" — This is useful when the current player skips over something or spends too little time on something. Good for getting them to slow down the action and include a bit of local colour, or to elaborate something they just introduced ("Describe that in detail. I want to know more about this burning village."), or to turn a quick thing into a more climactic scene ("…and the zombies are all killed." "Wait, wait. Describe that in detail. How exactly did we win with only five defenders, three shotgun shells, and a crippled doberman against a hundred zombies?")
"A follow-up scene, please."
- "I'd like an interlude after that." — These two phrases are ways to look at the aftermath of something that just happened. The first is asking the current player to do an additional scene to follow up what they just narrated (they can decline, unlike most of these phrases), while the second is a request to do a quick, one-minute follow-up of your own to add something cool that's relevant right now, outside of the normal turn sequence (which the group can decline).
At any time, the other players can also describe the environment or events around the character currently in the spotlight, but of course, these become subject to everyone else (including the current turn's player) using "Try a different way" and "Do it differently" to keep things in line.
(Of course, a player could choose to be difficult and constantly interrupt with "Do it differently!", but this isn't a game for players who don't work together and it doesn't complicate itself with rules-based controls to keep such players in line. It assumes that everyone is OK being responsible for keeping it enjoyable.)
There is one small random element in Archipelago, but it's optional and not a random decision mechanic. There is a deck of cards that can be drawn from to add random elements to scenes, but these are drawn voluntarily when you're looking for outside inspiration, so it's not a random resolution mechanic at all.
There's a decent Actual Play of a post-apocalyptic game of Archipelago here, and there's an audio recording of an Arthurian legends session here.
Two short form games
Love in the Time of Seið is a published game inspired by/based on Archipelago. The rules structure is very similar, but it takes it in an entirely different direciton by having five pre-made characters that have pre-existing relationships to each other during a tense time in a Viking-esque Kingdom's history. There's love, rebellion, betrayal, loyalty, and all kinds of conflicting tensions built right into the characters, but with enough of the details left blank (to be filled in by the players) that it has a lot of replayability, so each group's game (and each game by the same group) ends up differently. Unlike Archipelago, the game is close-ended: it goes until two characters are dead. I like to call it "Angsty Vikings" when describing it to people.
Montsegur 1244 is about a little-known but large branch of a European Christian heresy. All we know about them from history is their names—written down by the Inquisitors when the heretics surrendered at the end of the siege of Montsegur. Amazingly, half of them chose to burn at the stake instead of renouncing their beliefs. The game is about playing out the part of Montsegur's story that the historical record doesn't tell: what the experience of the long winter siege was like, and why these people chose the way they did after surrendering. The game has pre-set characters with short but tension-filled descriptions, and pre-set scenes with loaded setups but no prescriptions about how they should be resolved. After playing through the set scenes of the game, you have to decide what your character chooses in the context of what happened during the game's scenes, with the limit that one in the group must choose to burn instead of repent, and at most one can choose to escape.
Montsegur 1244, like Archipelago, has a random element in the form of prompt cards, but again they're an inspiration tool rather than a resolution mechanic: You might draw a card that says "Torches in the night", and that's only intended to put images in your head (along with the initial state of the current scene) to get creative juices flowing.
One bonus game
There is another such game, called Bhaloidam. It's statless, though not strictly number-free in that it has counters (which, being countable, are implicitly representing numbers), but they don't represent character stats as much as they do your/your character's narrative potency. By default it uses some custom d6, but it has a diceless play option. Like Archipelago it relies on the group to understand the characters and use them in character-appropriate ways rather than stat them up with numbers, and like Archipelago it allows others to control your character in certain system-limited ways. Rather than distributing GM powers, it has mechanics to say when each player gets to add to the action and how successful their character is in their in-the-moment goals.
The narrative power of the players ebbs and flows as the characters succeed or fail to influence each other and their environment. For example, character A might aim to stun B with a shield bash; if they succeed, A could move the "opportunity pawn" of B's player away from the "your turn" spot on the tracker, essentially moving back their "initiative" (to borrow D&D terms). Alternatively, A could choose to move some of B's players Will tokens to the spent pile, making them less narratively potent when B's turn comes up. (Or a little of both if their degree of success was enough.) Conversely, if A fails in their goal (stunning B with a shield bash), then they move their goal-invested Will counters to the spent pile and narrate something appropriate: perhaps B ducked the bash and got inside A's reach, setting B up in a good combat position (which will reflect B's player having their full narrative potentcy and being able to take their turn sooner rather than later).
What's clever about Bhaloidam is that the counters' positions — usable, spent, and some details about where you move them to commit them to goals that I haven't gone into) — and the "opporunity pawn" can represent whatever is important for your group's current story. In the above examples it was used to represent combat positioning and initiative, but they can as easily be used to indicate political favour in an intrigue-based game, magical potency and endurance in a game of planes-walking wizards, or sanity in a Cthulhu-mythos game. Its design is such that it does short stories or long, indefinite ones equally well.
The only "problem" with Bhaloidam (and why it's a bonus answer) is that it's still in production (development is done) and won't be out until next year. There's currently a Kickstarter to fund printing of the book and other components. (Of course I'm a backer already, for complete disclosure. There's a link to the Kickstarter on the game's page I linked above, so I won't directly link it here.) The news page I linked above has a bunch of posts about the game's mechanics, design goals and philosophy, play examples, and such.