I'm not aware of your dropbox syncing issue directly, but git does use hardlinks for local copies by default, and I can see how that could cause problems. Make sure to use the
--no-hardlinks option when doing a clone and you should be fine. Also, you can save account space by using
--bare for your dropbox repo and just pushing and pulling from there to another repo with a working directory.
Also, you may not necessarily need hosting at all. It's pretty easy to set up ssh on one of your own boxes. At work, I share a git repo with a colleague by just making it a shared folder on my computer. For just myself, I've even put a repo on a flash drive before, which can be faster for that first huge clone on a big project.