I've done this. I even wrote a blog post just like this. But I found out this doesn't work very well and in fact can easily leave you with a corrupted repositor

I've done this. I even wrote a blog post just like this. But I found out this doesn't work very well and in fact can easily leave you with a corrupted repository, specially if you try and push to the repository from a second machine. The problem is aggravated because git creates a lot of files and Dropbox doesn't have rollback by directory, only file by file. There are many blog posts and answers over the net by people that realized this just doesn't work so well. I currently use bitbucket for free private git hosting. I also have a script that uses git bundle to create an archive inside my Dropbox folder, just for redundancy.