using cloud space as a private git repo

It is absolutely possible – what you probably want to do is create a “bare” git repository in folder Y (git init --bare)

and then add that file location as a remote:

git remote add Y file:///path/to/Y
#example path: file:///media/sf_OneDrive/gitrepo/FenicsSolver/

git remote set-url origin --push --add <a remote>
git remote set-url origin --push --add <another remote>

The latest version of git (as of October 2012) allows you to do this from the command line:

git remote set-url origin --push --add user1@repo1
git remote set-url origin --push --add user2@repo2
git remote -v

It is also possible to push encrypted code to public repo, not tested.