The error you see is indeed because the allocated space for local storage has been exceeded (on Chrome in this case). The only thing that the client stores in local storage is the session data, which is very small. It's also overwritten every time the client gets new session data (which is on reload) so you won't run out of space.
It is possible that you're using custom widgets which write data to local storage. If you could file a ticket with an example application in which we can reproduce it, we'll take a look at it.
Thank you for reporting this.