Private social media is a bit of an oxymoron. People on social media don’t REALLY want privacy. If they wanted privacy, they wouldn’t be on social media.

They want an engine that will go through their pictures, posts, etc. determine what other users might be interested in seeing their posts and to be connected with those users in an effort to increase their influence, grow their “friends” list, etc.
I’m MOST certain that it can be built. I’m also sure that there are folks willing to take your money to build it for you.

But, $1 a year won’t even come close to providing the funds required to continuously maintain the security of the site as new exploits are discovered. And, since you’re not on AWS or Google, you’re going to have to pay for a team capable of keeping a the systems up, hardware firmware patched, routers protected, (plus multiple replicated sites to avoid outages) and available for potentially millions of customers. And, that’s even before you get into the effort of your teams that would be required to moderate the service. And, with ZERO tracking, it would be very difficult to have functional groups (as, on the back end, you’d need to track what groups they belong to).
If the idea is that it runs at a loss because you’ve got enough personal money to keep it going, then it could be a good service for a lot of folks that ONLY want to share with a known group of people. As others have mentioned, though, they’re already doing that for free elsewhere, so they would need an incentive to migrate. The cost I don’t think would be a big problem as the “money haver’s” in the folks that want to share likely wouldn’t mind paying $20 for spots for themselves and 19 others.