Are you using Safari? If so, that's likely the problem. See here.
Workarounds include not using Safari for Twine development, which sounds dismissive and lame, but it also solves a few other potential issues. Safari is stricter than it needs to be with just about everything from media to web storage to saving files.
You can also try saving the page that comes up as something like "story.html" (File > Save). This usually works. You can also create a new text file somewhere and name it something like "story.html" and then copy/paste the code from the browser to the text file. Either method should work.
Again though, if it's possible to install a different browser on all the affected machines, that might be the most cost effective solution.
If you aren't using Safari, please let us know what you are using, as this may be an exciting new bug.