I don't know about SVN, but you can definitely set up a Git repository without needing to install Git on the remote machine. And Git > SVN :-). It's a bit complicated though.
Edit: I tried to find a walkthrough to do this but failed as yet. But the basic steps are pretty simple (this is all from memory from a year or two back though):
- Initialize a bare git repo on your local machine. This will act as the origin to the only person who would be able to push to the remote: you.
- Create a trigger that copies the contents of that directory to the remote server (FTP, rsync: whatever's available to you) on commits, etc.
- For you, clone the origin located on your local machine so that somebody can push to it. For all others, they should clone the directory on the remote server as if it were an "authentic" git repository.
- Configure access rights accordingly, however you can.
It's not a "true" Git repository I suppose, but I've used it before and it works for the purposes I used it for.
The Git User's Manual goes through part of the steps involved.