What story format are you using (I'm assuming Sugarcane)? What browser are you testing with? Did you try building to see if the problem persists? How about another browser? Do you have any custom JavaScript code running? Were you able to play after the error as the message suggested? If so, how'd that go?
At a guess, this is probably an issue with Safari, which has changed the way browser storage works on recent versions, wreaking havoc with a lot of web apps. The Twine formats that are still actively maintained have all been updated to deal with this, but no one is currently maintaining the Twine 1 default formats.
For this reason, it is highly, highly recommended that you use SugarCube 2 if you want to keep working in Twine 1.x.