You can reproduce this by setting Block sites from setting any data in the Chrome/Chromium Content Settings.
Visiting en.wikipedia.org you will get:
load.php?debug=false&lang=en&modules=jquery%2Cmediawiki&only=scripts&skin=vector&version=0f3xo9s:176 SecurityError: Failed to read the 'localStorage' property from 'Window': Access is denied for this document. DOMException: Failed to read the 'localStorage' property from 'Window': Access is denied for this document.
at Error (native)
at Object.init (https://en.wikipedia.org/w/load.php?debug=false&lang=en&modules=jquery%2Cmediawiki&only=scripts&skin=vector&version=0f3xo9s:171:803)
at Object.work (https://en.wikipedia.org/w/load.php?debug=false&lang=en&modules=jquery%2Cmediawiki&only=scripts&skin=vector&version=0f3xo9s:166:656)
at enqueue (https://en.wikipedia.org/w/load.php?debug=false&lang=en&modules=jquery%2Cmediawiki&only=scripts&skin=vector&version=0f3xo9s:163:476)
at Object.using (https://en.wikipedia.org/w/load.php?debug=false&lang=en&modules=jquery%2Cmediawiki&only=scripts&skin=vector&version=0f3xo9s:169:277)
at runScript (https://en.wikipedia.org/w/load.php?debug=false&lang=en&modules=jquery%2Cmediawiki&only=scripts&skin=vector&version=0f3xo9s:161:350)
at checkCssHandles (https://en.wikipedia.org/w/load.php?debug=false&lang=en&modules=jquery%2Cmediawiki&only=scripts&skin=vector&version=0f3xo9s:162:124)
at execute (https://en.wikipedia.org/w/load.php?debug=false&lang=en&modules=jquery%2Cmediawiki&only=scripts&skin=vector&version=0f3xo9s:162:826)
at Object.implement (https://en.wikipedia.org/w/load.php?debug=false&lang=en&modules=jquery%2Cmediawiki&only=scripts&skin=vector&version=0f3xo9s:168:843)
at https://en.wikipedia.org/wiki/Main_Page:7:2796logError @ load.php?debug=false&lang=en&modules=jquery%2Cmediawiki&only=scripts&skin=vector&version=0f3xo9s:176.Also login wont work getting you an similar error and the message
There seems to be a problem with your login session; this action has been canceled as a precaution against session hijacking. Go back to the previous page, reload that page and then try again.
Looking at the implementation I assume that setting localStorage: window.localStorage, as a field might be a problem if window.localStorage does not exist. Are there any plans to improve mw.storage? E.g. by using jquery.jStorage, that seems to give a much more robust implementation for storing local data.