The simplest (really the simplest) would be to do a git init --bare in a directory on one machine, and that way you can push or pull from it, with the directory path as URL from the same machine and using ssh from the other (you could do this bare repo inside a container but really would be complicating it), you would have to init a new bare repo per project in a new directory.

If a self-hosted server meaning something with a web UI to handle multiple repositories with pull requests, issues, etc. like your own local Github/Gitlab. The answer is forgejo (this link has the instructions to deploy with docker), and if you want to see how that looks like there is an online public instance called codeberg where the forgejo code is hosted, alongside other projects.