So WebStorage of HTML5 acts very much like cookies, but if more flexible. It does persist like Ashley said after all browsers are closed and after the computer has been turned off. But its downside is IF a user clears their cookies AND unchecked (in IE) the Preserve Favorites and Websites data checkbox it will clear the WebStorage.
You cannot write to XML or other files for long term storage because as someone has previously said, that is a limit imposed by the Browser and is a security violation.
So posting to a website/URL and that triggering a save to an online database is the only option I can think of at this time. You can also allow players to play offline and when they are back online have it save to the web those changes, but not sure how that would be coded in C2.
The ideal long term solution would be for C2 to enable support for SQL Lite databases on the local computer much like Apps do on Mobile.