Because these are toys, I don't need security, backups, scale, or size. Which makes almost any choice I professionally know about feel like overkill. I do not want to have to worry about or maintain a web server.
I feel like I am looking for a service something like this:
1. Make an account and receive a token.
2. Embed the token in my game.
3. Use the token to make GET and POST to the service.
4. Don't abuse it or my token gets rescinded and now nobody can access their high scores or whatnot.
But more in depth, it's a good question!
You could get innovative and use localstorage combined with WebRTC to create a p2p storage "mesh". :)
Honestly tho I would use GitHub gists. They have an API. There's a token you can get. You could use whatever storage format you want (JSON anyone? Blah, why not just use malbolge).
Or if you're doing gists, why not go the whole nine gits? Depending on what your game runs on, you could just use git, to a shared repository, on GitHub.