Iframe in Chrome error: Uncaught SecurityError: Failed to read the 'sessionStorage' property from 'Window'

I had the same problem with localStorage, and fixed it like this :

Under Settings > Privacy > Content settings, change the cookies' settings to "Allow local data to be set" or the second option (in my case, it was previously on "Block sites from setting any data").

Edit

On newer Chrome versions, make sure the first option is ticked here: chrome://settings/content/cookies

I had the same issue, but my problem was a script blocking plug-in ScriptBlock that I had set to allow the site but was still blocking some actions and not reporting it.

Disabled the script blocker and voila!