I cannot find any official details on the Far Realm for 5th edition Dungeons & Dragons. Is there any published lore on the topic or is it supposed to be open ended?
According to Dungeon Master's Guide p.68, conveniently available on the Wizards of the Coast's website as a PDF:
The Far Realm is outside the known multiverse. In fact, it might be an entirely separate universe with its own physical and magical laws. Where stray energies from the Far Realm leak onto another plane, matter is warped into alien shapes that defy understandable geometry and biology. Aberrations such as mind flayers and beholders are either from this plane or shaped by its strange influence.
The entities that abide in the Far Realm itself are too alien for a normal mind to accept without strain. Titanic creatures swim through nothingness there, and unspeakable things whisper awful truths to those who dare listen. For mortals, knowledge of the Far Realm is a struggle of the mind to overcome the boundaries of matter, space, and sanity. Some warlocks embrace this struggle by forming pacts with entities there. Anyone who has seen the Far Realm mutters about eyes, tentacles, and horror.
The Far Realm has no well-known portals, or at least none that are still viable. Ancient elves once opened a vast portal to the Far Realm within a mountain called Firestorm Peak, but their civilization imploded in bloody terror and the portal’s location—even its home world—is long forgotten. Lost portals might still exist, marked by an alien magic that mutates the area around them.
Player's Handbook, p.302, summarizes the same information.
Seeing a creature from the Far Realm can risk an individual's sanity (DMG p. 265).
Githzerai philosophers are aware of the existence of the Far Realm, and it may be connected to the aboleths somehow (MM p. 14).
Some users of wild magic are so afflicted due to contact with the Far Realm. (PHB p. 103).
Some warlocks use the ancient knowledge of beings of the Far Realm. Warlocks of the Great Old One may worship such unfathomable beings. (PHB p. 105).
Powerful, world-shattering beings known as the Elder Evils are speculated to be creatures of the Far Realm (Mordenkainen's Tome of Foes p. 234).
Far Realm lore from earlier editions of the game can also be used in your campaign. However, much about this plane is intentionally left undefined in order to set a mysterious and unsettling atmosphere, so lacking much rigidly defined lore may be to your advantage. You may find it useful to read the works of H.P. Lovecraft for inspiration.
The Far Realm was named and invented in 2E AD&D in the adventure "Gates of Firestorm Peak" written by Bruce Cordell in 1996, a mountain home of some Duerger dwarves had a tear in reality that leaked bizarre creatures into our universe.
Far Realms was officially set in D&D as homage to H.P. Lovecraft fiction and similar mind and body warping alien influences: think Cthulu mythos, H.R. Giger's ALIENS and weird biomechs, the Thing, and Salvador Dalí weird paintings, that eye in the hand monster "Pale Man" from Pan's Labyrinth, the Blob, microbial/amoeba-like life, tartigrades, cephalopods and jellyfish, and at the very least the creatures from Stranger Things; although there is some dispute as to if the Upside-down is analogous to the Shadowfell introduced in 4th edition retconned D&D cosmology, or a better fit in 2nd ed. Far Realm of the Great Wheel Cosmology.
No mention was made of it in the original Manual of the Planes (AD&D 1st ed.) or Planescape setting (AD&D 2nd ed.) As TSR had been bought by Wizards of the Coast in 1997 and many things on the table were scrapped or wrapped up, and many things brand new or previously off limits were hurriedly put into print at the end of 2nd edition's life span such as Far Realm and the Player's options before the final switch from TSR to WotC. several of the Far Realm monsters made it into the Monstrous Compendium Annual Volume Four accessory published in 1998 near the end of 2nd ed.
Several classic old school D&D monsters that were strange, gooey, weird or downright alien were assumed to have originated or at least been corrupted by the Far realm such as : Beholders, Flumphs, Mind Flayers, Gibberlings, Aboleth, Grell, Fehr, Gibbering Mouthers etc. If it had tentacles or inflicted madness it probably originated there. The abberation template of later editions of D&D was a good indicator of what kind of beasties might have originated there.
The idea of a "far realm" was certainly there from the very beginnings of D&D with spells such as "contact other/higher plane" allowing mortals to contact strange and unusual powerful beings from some nether-place far away on a different plane of experience usually with drastic consequences that usually led to madness, death, slavery or mutation, but for a lucky few dark otherwise unattainable powerful knowledge. It was later postulated that Psionics (introduced in OD&D'S Eldritch Wizardry way back in 1976) was a way to combat madness and entities from the Far Realm.
Mr. Cordell and friends continued to mention and expand the Far Realm in several 3rd edition products explaining that it consisted of many transparent layers from inches to miles thick of strange viscous fluids that were stacked together that still could be seen through. Multiple layers were visible at once like transparencies on a projector obscuring the farther/deeper one peered, with huge beings that crossed multiple layers at once a reality-defying madness-inducing concept for creatures from just one plane of existence. Many beings there were described physically as somewhat reminiscent of deep sea creatures but with more eyes, mouths and tentacles, and as old or older than the gods themselves.
The Far realm multiple parallel layers concept is not that dissimilar to branes in a multiverse configuration of string or M Theory of parallel dimensions in real world theoretical astrophysics. There are a handful of Far realm mentions in 4th edition and 5th edition Dungeons & Dragons, mostly in planar information sources or as explanations for abberations.
Other than the few paragraphs in the DMG (p.68), there really isn't much, no. It's intentionally vague, both because it's a playground for a DM, and because it's meant to be essentially unknowable, a place beyond human comprehension.
There's slightly more in previous editions, and some people have done DM's Guild updates of old adventures that center around the Far Realm (notably "The Gates of Firestorm Peak", which the DMG alludes to), but even then the basic nature of the far realm is "that which man was not meant to know", so it has tended to stay pretty well unexplored throughout D&D's existence.
Not much is known, but from what I can gather, there are some wizards called the kaortu who got mutated, some old gods and a couple of giant lovecraft-squid thingies. For laughs, you can have some have a mindset like a cats, or unable to comprehend simple things like "shapes" but understand the fundamentals of the universe and spout out the quantum mechanics of magic while asking what is "color". The rulers, who live like the others on top of said giant squid thingies are the uvuudam, wizards with spear-tentacles for heads and are similar in power to demon lords. Do not screw around with them. Utilizing dimensions like time differently than ours, we get transformed to better accomodate the concepts of perceiving more than four dimensions (the amount most humans can see, I guess) and go mad from it. Or maybe saner than before. After all, it may be the truth......