GitFS
I wondered about this a while ago, and now GitFS exists (via Andreas):
gitfs is a FUSE file system that fully integrates with git. You can mount a remote repository's branch locally, and any subsequent changes made to the files will be automatically committed to the remote.
gitfs was designed to bring the full powers of git to everyone, no matter how little they know about versioning. A user can mount any repository and all the his changes will be automatically converted into commits. gitfs will also expose the history of the branch you're currently working on by simulating snapshots of every commit.
