It seems you would want a hosted git installation like github / gitlab or VSO for that. The main concern to me is that API keys are usually too limiting (or too expensive). E.g. if you don't want to manage local state by having a working directory you need to do many API calls to make a change (create the objects, create the trees, then create the commit, then change the ref) so that it is almost more worthwhile to use the same data structure on a more generic backend like a NoSQL key-value store. I haven't done this though (though it's been on my wish list for long).

What it also doesn't give you for free is sensible search indexes. I do think though that combining it with a search index could be very powerful.