It is indeed stored on an Edsu server. The data format is trivial and completely specified, so switching providers should be straightforward (an Edsu app to do the data transfer would be easy to write, and I plan on writing one). If you're using your own domain (which is encouraged), from there you'd just update an A record to complete the transition, if not, you'd set up a transparent redirect on your old host to keep all your old links/permissions alive.
It's pretty old school in that if your provider loses a server and they're not doing replication/backups then there can be data loss. If they are, then a failover should work fine, with minimal loss and no corruption (the important stuff is atomic).
With localStorage, how it's used is up to the app (generally it'd only be used for keeping a token and maybe a block cache), so I wouldn't expect a loss of that to matter in most cases.