All browsers, that I know of, treat localstorage as part of your profile and store this inside your home directory somewhere - which is where all your other stuff is, so if you're not backing that up... I can't help you.
For example, Chrome stores it here:
on Windows: %LocalAppData%\Google\Chrome\User Data\Default\Local Storage
on Linux: ~/.config/google-chrome/Default/Local Storage/
For example, Chrome stores it here:
on Windows: %LocalAppData%\Google\Chrome\User Data\Default\Local Storage on Linux: ~/.config/google-chrome/Default/Local Storage/