Joseph Sisko's restaurant is his personal property, not his private property since it is not a money-making venture. Since money, and capitalism, do not exist in the Federation, there is no private property in any form. Furthermore, given Star Trek's egalitarian/utopian vision of the future, no one is going to take Joseph Sisko's restaurant -- the laws of the United Earth government (which has direct jurisdiction over Earth) exist (imo) to protect people's personal property, not take it away.

The government can seize private property in America via eminent domain. It's not a question of seizure, but what the space is actually being used for. Again, this is largely an economic question, so you can't sidestep economics.

It is interesting to consider that in the vastness of space that something like a single restaurant might be viewed similarly to a glass of water in the US.

Sure the government could come in and declare eminent domain on my glass of water, but it’s value is so low as to be effectively a nonissue.

In a future where there are tons of planets and tons of replicators, perhaps the idea of personal property has just been extended to include things like a restaurant or a vineyard.

If you use the definition that private property is the private ownership over the means of production, it could be reasoned that something like Sisko’s is not necessarily a means of production but more akin to personal property. If someone on earth wants some creole food they can use any number of replicators to produce and enjoy that. Sisko’s and Picard’s vineyard might be similar to how we would look upon historical preservation. Some people could choose to spend their lives making things the old fashioned way because they enjoy it and people enjoy experiencing it.

The economy of Star Trek is interesting, but I think there are plenty of times when the utility of storytelling ends up mucking with the clarity of the message. One example I was just thinking about the other day was the introduction of the borg queen.

I get why it’s nice for there to be a borg queen, she can embody a more nuanced thinking part of the borg collective and the audience can much more readily understand the idea of a queen ruling over her subjects (whether that be like the rulers of humanity or like the queen bee as they sometimes say). But it also kind of sucks. The borg are such a fascinating species, a collective hive mind acting to attain perfection, more a force of nature than any of the other species we encounter.

While the borg queen is a compelling character and is acted wonderfully, I can’t feel a bit sad that it’s so normal and pedestrian. It turns the borg from this almost incomprehensible force into something so regular, a bunch of drones carrying out the will of the queen. While expedient to the storytelling, I like the idea of what the borg are pre-borg-queen more than what they become post-borg-queen.

I think with the economy a similar thing happens in storylines. There are many scenes that make it clear that humanity doesn’t have money anymore, but when you are telling a story and you want to have some stakes and obstacles, money is soooooo useful. Money makes it trivial to have an obstacle, or shit we need some latinum. Money makes it trivial to introduce stakes.

Star Trek had to try to thread this needle of presenting a post scarcity society while also making a dramatic engaging show for people living in a capitalist society. Scarcity is at the heart of a lot of drama, if you can just replicate your way out of every problem it’s not a very interesting show. It also leads to a thing that once you spot it’s hard not to spot, so much of the tension is aided by the “oh no we can’t replicate that” McGuffin. It plays out in a lot of episodes because otherwise every episode would be 5 minutes of “there’s an outbreak of tallarian flu on Corso V, we emailed them the recipe for the medicine and told them to replicate it.” Then the credits roll.