Chrome extensions/apps have sync storage options that are provided for free with a Google account. But not a centralized service to collect data (if you do not take Google Analytics into account).
Sticking with Google ecosystem, though, you might look into
Google Apps Engine.
There's a certain free quota and your usage might fit into that. Then again, it might not, but worth a look.
Here's a few links on the topic:
SO: How to implement internet high scores in Google App Engine
Google blogs: Building scalable social games on App Engine
Google blogs: Google Code Jam's Ranking Library
Github: Pre-made GAE server "omgleaderboards"