So all of a sudden, my Hangouts window was prompting me to sign in. I clicked the button, it opened a browser window asking for my password. I logged in, said window closed, the Hangouts window refreshed… and asked me to sign in again. I was stuck with this, and it happened on all three computers I am using Chrome (and the Hangouts extension) on.

Here’s what solved the problem for me:

• opened Chrome settings -> privacy -> content settings
• In that window, clicked the “manage exceptions” button in the “cookies” section
• added an exception (allow) for https://talkgadget.google.com

Now after one more log in, it started working again.

Edit 2015-09-15: Apparently Google knows about this issue and added a hint to the Hangouts sign in window: