I am having this issue with all accounts through mail as well as through iCloud. All gmail accounts prompt me to enter my password. When I enter the correct password, the screen goes gray and does nothing. Outlook prompts a different error:
"Connections to host outlook.office365.com on the default ports failed."
These issues appear to have started randomly on Friday but maybe there was a small update I am not remembering.
More concerning, perhaps, is the fact that my computer also prevents me from logging in to iCloud. Similar phenomenon - when I enter the correct password, the verification applet becomes unresponsive or says "An unknown error occurred".
The big problem here is that now I can't access the app store to receive updates!
I see that some people have had luck with removing and re-adding accounts. This is not a fix! It's a work-around. So...where is the fix?