Chrome storage would be the default, and can be used to sync information between computers for a logged in user.
https://developer.chrome.com/extensions/storage[
^]Bear in mind too that chrome storage does have data limits, I think 5MB for local storage and 100KB for sync storage.
Aside from that, i think you're looking at needing something you can access with an API to store data, maybe in Google Drive, or using an App Engine backend, or Parse.com. perhaps.