If you want something rare or precious (such as land for a farm) presumably you can apply for it in a lottery or to a commission intended to apportion such things equitably. (Source: an assertion in another SFF answer).
Let's restrict this to the main topic: Land to the Peasants.
Is there canon support for this? If no, what is canon evidence for how an average Earthling (in Picard's time) would be able to get a land plot of the size they want, in the location satisfying their wants?
What a wondrous question. I'm aware of nothing in canon that will answer this question with any satisfaction for a very good reason: it would limit the story possibilities in a profound way. Fiction isn't meant to answer every possible question of some utopian future but rather be something to aspire to, a dream.
So with that in mind, let the speculation begin!
Land ownership within the United States is fraught. There are many rules that apply, from zoning rules that restrict usage, to the requirement for public access easements, to the eminent domain power of government to "claim" property for the public.
It's important to recognize these caveats because even in modern society ownership has its limits. The Sovereign, the people of the United States, in our case, grants certain property rights but they are not limitless or irrevocable.
Things have changed a lot since the King granted Lords their land titles but there is still ownership, it seems like there's plenty of room for things to change more but the concept of ownership could still exist.
One important note: Land Ownership has often been important in the past and modern day because it serves as an extremely stable means of earning income. From Lords allowing serfs to work the land and taxing them appropriately to restaurants and developers making a profit off owning highly desirable land.
Without money, land ownership wouldn't be motivated by generating incoming, it would be tied very strongly to particular uses and other motivations.
It's also worth noting that in today's era property rules tend to be much stricter in high density locations or in areas that are desirable. Let's use the example of the being near the beach!
On the East Coast it seems much more common to have "private beaches," an owner can own that land right up to where the water touches it. However, on the Best (West) Coast the California Constitution states that no entity can claim to own the beach, it "belongs" to the people of California. (CA Const, Article X, Section 4)
Zoning laws are often used by government entities to essentially restrict how land gets used (and, indirectly, who wants to "own" the land) and the National Parks system explicitly establishes land to be preserved for the people.
Two important points to our analysis: (1) Earth is not overcrowded. Whether we are visiting Starfleet Academy, San Francisco, Paris, vineyard, Harvard, or any other Earth location we don't see any evidence of severe overpopulation, buildings are not especially high density, there are examples of spacious homes and small bars. Even Barclay's apartment in San Francisco (VOY: Pathfinder), though in a high rise, is exceptionally spacious for a bachelorpad. (2) Views of Earth from above confirm a staggering amount of "green space" that is not developed.
So however "ownership" is handled, there does not appear to be a major scarcity of available land.
If anything, if we assume that birth rates in the Trek era are closer to advanced society's of today (using the term "advanced" loosely), then the population would remain very stable (and probably small) and there would be a surplus of land.
Not to mention the ability of the human population to leave the mothership (Earth) and go explore might mean a much reduced population (at least rural population, cities might be even denser than today but transporters would make it easy to escape the tedium of city life).
We don't get to see many examples of land ownership within the Trek series. Kirk "owns" his house in Star Trek Generations, Picard's family vineyard (Sikso's restaurant ownership might not actually include the land) so it's difficult to know what "ownership" means. Nor can we really use personal property to indicate what the rules of land ownership would be. Land ownership has always had it's own rules and regulations within Earth society and it is reasonable to assume that continues into the future.
So what are some possibilities that would allow for our very limited examples of ownership (a house and a vineyard)?
First, it seems likely, as it is today, different rules would apply to different locations.
If you want to build a cabin in the woods you might merely need to establish that you actually intend to live there, that you really do move-in and that it won't just sit there falling apart. The Federation could easily grant you a deed for use.
Second, actual demonstrations of usage might be a requirement. Abandon your land and it may be subject to being claimed by others.
Third, it's not purchased or sold for money (it's difficult to make the case for any sort of consideration because it seems likely it would become a form of money).
Fourth, because land ownership comes with added responsibilities and doesn't grant the right of exploitation it might not be as highly desirable as it is today.
Picard's vineyard is a great example of why not having money would be very important -- without the money the only reason to run a vineyard is the pleasure of making wine. There would be no profit in making wine and selling it. So it seems likely that to "own" the land they must "use" the land. Picard's brother, Robert Picard, was not to happy with his choice to join Starfleet because it meant he had to remain and maintain the family vineyard.
Since the family would be comfortable regardless of whether they owned the vineyard, perhaps the key to ownership was, in fact, that the land was being used by their family? Perhaps some of Picard's grief in Star Trek Generations was motivated the idea that the family might lose the vineyard?
Today ownership endows the owner with the right to exploit the land for profit and prevent others for utilizing it for anything else. There's no money, so the idea of profit (at least in the sense we understand it) would be lost. But what about deciding the "fate" or "use" of the land (e.g. "I can walk, build, swim here, you can't!")?
If that is the main right of "ownership" in Star Trek's universe then you could easily imagine tying certain rules and responsibilities tied to such a right. Clearly the government can't "tax" the land but what about other use stipulations zoning, access or upkeep requirements?
In any case, the loss of exploitation for monetary profit, would probably decrease the number of land transactions (I'm purposely avoiding the idea of "condominiums or apartments" as it isn't pure land ownership, though clearly one wonders how highly coveted land would be handled).
One thing that might help alleviate the concern of pure ownership as we understand it in the 21st century is that "sharing" might be even more convenient and accepted given transporter and replicator technology.
This still leaves a great deal unanswered. What about institutions, like Starfleet Academy, that have a need to expand their campus? What if there were a population boom, an influx of refugees from a planetary disaster? If you want to live in isolation, how do you obtain enough property to do so (or do you just have to leave)? What about people who reject technology?
More to the point of your question, how could some random human, born in space with no family lands, get their own plot of land?
I'd argue they'd choose their plot of land, see if it's available and then go file a request for ownership and use at a local land office. They could request to see if any property owners were selling their land or claim an unused plot so long as they met certain requirements for use and occupation much in the same way the western states had a homestead system.
As mentioned above, because it appears to me there would be limits on the amount of land a single entity could claim and use but with a stabilized population and a big galaxy I'd say they could claim land with relative ease.
You might imagine there would still be strong local governments to control and disperse the land in their dominion. These governments would be empowered to set their own rules and guidelines for staking and maintaining a claim.
Ultimately, this is all pure speculation but my point wasn't to put forth the Star Trek way but merely use the limited evidence we do have to infer some possibilities about how land ownership could be handled in a society with no money and no concern for material wealth.
Fun times, I do hope this provides some insight. If I note or think of anything else, I may add it.
As the economy is merit-based, you would have to prove first that you have something to offer for the society. Widely known starship captain upon retirement would have much higher chance to receive that plot of land than young person. Other way to do that would be "trust-credit". You have been given land and show us what you can do with it and how we all benefit from it.
Money is a way to gauge the value of scarce (finite) resources everyone needs. But it's not simple: five basic things every human being needs to LIVE are usually free: air, water, sunlight, roof and food. How do you put a value on something like that? On related note: if there is no need to compete for electricity to light and heat 120-bedroom mansion (as they are no longer a thing anyone craves) what would be that power's value? And the mansion's?
EDIT: Source canonical supporting (indirectly, true) above, two best: "Star Trek: First Contact".
"The economics of the future is somewhat different. You see, money doesn't exist in the 24th century... The acquisition of wealth is no longer the driving force in our lives. We work to better ourselves and the rest of Humanity."
Second is "TNG: Time's Arrow pt2"
Samuel Clemens: Young lady, I come from a time when men achieve power and wealth by standing on the backs of the poor, where prejudice and intolerance are commonplace and power is an end unto itself. And you're telling me that isn't how it is anymore?
Counselor Deanna Troi: That's right.
Obviously, I can't point to canon support of everything I wrote, as it is an extrapolation, (un)fortunately, however based on the multitude of single statements, sometimes even only partials. But I stand by my analysis. Something had to replace what Clemens enumerates. Humanity is not wired for "work for the sake of work". Had it been so Communism would triumph. And in ST merit carries in Federation as a whole, as can be seen all too often.
EDIT 2: Also, what would be the explanation of existence of Picard winery, for example? Or Sisko Senior's restaurant? Or multitude of other small things?
EDIT 3: How about advancement in Starfleet? What is the process of obtaining single/better quarters, especially aboard starship?
