It is now 2015, and as of three days ago, a new tool based on Dropbox API v2 has been created to safely use git on Dropbox. It works against the API rather than using the desktop client, and correctly handles multiple simultaneous pushes to a repository hosted in a shared folder.
Once configured, it allows one to set up a git remote exactly like any other git remote.
git clone "dropbox::/path/to/repo"
git remote add origin "dropbox::/path/to/repo"
