The only solution was to use Chrome... which worked until very recently. Now I get the same problem with Chrome too, albeit the timeout is much longer, perhaps an hour or so or sometimes longer.
Everything is fucked.
By the way, this is Apple bug 29018740. It might help if someone makes another bug report and references this bug, though I doubt it because when the latest Safari version was released (with the bug still present, of course), I mentioned this to Apple and they responded:
> Thank you for contacting us. If this is still an issue for you on current releases, please file a new bug report.
By the way, this is just for text messaging. For voice/video Hangouts became unusable earlier than that. The video quality is approximately 120p and the audio is 1kbps or less, and I can't talk because noise cancellation doesn't work and I hear myself back with a 2 second delay and 80dB gain. Don't even get me started on CPU utilization yet...