Picard doesn't count, what with his family's fancy-shmancy chateau and vineyard

My head canon is that instead of monetary rewards for doing important or dangerous work, you get more land to use. That would mean trashmen, farmers, nurses, and starship crew would have huge palaces and ceos would live in a shoebox. Picard gets double sized because he's a captain and a farmer. Same with Boimler.

Poor Boimy has it rough.
I rather thought that there were no “palaces” and no “shoeboxes“, but still a varied range of accommodations that are a mix between inheritance and state ownership where they lend it to an individual or our family.
There’s a very good reason why most sci-fi series that espouse a futuristic socialistic utopian future never really get into the details. Because we, today, haven’t figured that shit out yet.

If you had teleportation available for transport, wouldn't you move into the middle of nowhere?

After vetting local flora and fauna.
- Dinner! (For a ~~month~~ year!)
- Have you seen/read MurderBot ?

My guess is that in post scarcity societies, a cabin in a remote location far from everyone is the easiest thing to get. There's plenty of empty land all over the world.

City centre on the other hand, apartments might be small with a waiting list.

The entirety of the Maquis rebellion, explained in light pollution maps
About the only way the apartments the cast from Friends live in could make sense.

This begs the question: who gets the beachfront property on Star Trek Earth?

The Commodore
- Man this episode makes me cringe. Although the follow-up is arguably worse, with Geordi getting all indignant that someone would be upset at him dating a holographic deepfake of them. Is it so hard to write someone being socially awkward without making them a total creep?
The answer, whomever wants it. Since most everyone has easy access to anywhere they want to go and anything they want to do, things we consider luxuries are trivial pass times.
- And what if there's more demand than supply?

We even know Sisko was planning on building a cabin in the woods on Bajor over looking some river. He bought land sometime in the later seasons.

in a post scarcity society with industrial scale matter replication and matter/antimatter power sources—why the hell not? I’m sure anyone could have one if they asked.

When you control large portions of the galaxy handing out land is like handing out stock before an IPO

Doesn’t everyone want a cabin in the woods?

Sure, but not the one run by these guys:
- I found their revised black board:

It’s my belief that Starfleet still pays their officers, but that since they spend so much time on missions, they have fewer opportunities to spend the money. And by the time they become captains, they’ve probably saved up quite the nest egg.

How do you think Sisko afforded that land on Bajor? Though, being Emissary might have afforded him free or deeply discounted land.

Bargain, for sure. I wonder what Kasidy's doing with that plot these days. Let's check in...
DS9 is probably the closest we got to really investigating federation economics. At some point Sisko warns Quark that if he stops being on the Federation's good side, hell make him start paying for rent and utilities again. So they have some concept of converting "time and effort" into "credit". The star fleet officers all have latinum to spend at Quarks, and Quark complains on several occasions that business gets slow when Starfleet stops visiting his bar, so he's definitely getting paid.
My assumption is that officers are given an allowance or extended credit for interactions with other cultures, and that allowance/credit is either a higher number or simply "more lax" the higher rank you are. Want to "expense" a holosuite visit at Quarks with Starfleet, you'll have to explain yourself less the higher rank you are. Kind of like "unlimited PTO" at big tech companies.
Like, an ensign might request a cabin in the woods from the cabin logistics department, but they'll be in line behind all the captains, whereas an admiral would be able to jump the line.
- This question came up a lot in DaystromInstitute.
  The general headcanon is that rent, access to the stations grid and crew for maintenance is credited at fair market value for officers to use at Quarks. Plus having a steady stream of Starfleet or Bajoran officers there would deter general crime and ruckus, much like why gas stations tend to offer free coffee for cops.

Maybe because as a captain they spend so much time in a starship that they have need to live in a large open land with a wood/stone cabin.

Like with developers, we end up looking for hobbies that doesn’t involve technology.

Post Scarcity Society have a cabin in the wood for everyone

Honestly with replicators and the amount of forest that grows naturally on Earth, we could have a lot of people do this.
- Think about how much land area is currently used for growing crops. If the remaining farms are small vanity/tourist places like Picard's, then millions of acres on the planet would be rewilded.
  Having a cabin in the woods could be quite common.

Aesthetically, it works extremely well because it contrasts with the Enterprise. Anyone who is a fan will feel on some level that the Captain is somewhat out of their element when they're not on board the Enterprise.

Also lots of nautical references.

Isn't the Kirk image from Kirk Heaven?

Also, he achieved the rank of admiral prior to this scene, and self-demoted. So, I mean, Kirk, at least, makes some sense. He also had the smallest, most reasonably sized, cabin.

This is a shot from the Nexus, but the cabin itself he did previously inhabit. He describes in hilarious ADHD manner various memories as Picard tries to reign him in.
he wasn’t self demoted, he was demoted to the rank of captain as punishment for stealing and then losing the enterprise, but got a dispensation for saving Earth from the chatty whale probe.