Git works through ssh. So you need the same system as sshing into your machine. You just make a user group git and then let git and ssh handle things. And if you don't need people to push to your repo, then it's a lot easier as it's now similar to hosting a website/file server.

Git exposes a lot of internals through odd commands, so I suspect you could manage synchronization by sending changes over email or something.

Bonus fun fact: there's a git bundle command that "dumps" the repository into a single file, that can be interacted with as a remote. So if you're ever working with a local repository and want to put it on a server over ssh or something like that, you can just create a bundle, scp it over, and clone from that on the server.