Got this today as well, after updating to 10.14.6. I have both personal as well as business related emails via Google, and none of them work. Same with iCal. Going to Safari, literally, does nothing. Tried also via all Google accounts, passwords, settings -nothing.
Tried:
- removed keychain history - didn't help (won't let reenter passwords on Mail, won't enable, won't take online)
- going to google account security and turning on less secure apps - didn't help (was recommended at google forum)
- removing/re-adding the account to Mail - didn't help (won't let adding google-related account back; also not via "other")
- disconnecting iCloud account and re-connecting - didn't help
All calendars of course off, as well. Using emails via browser at the moment and downloaded Outlook (a pain!).
HELP, Apple? This seems to be reoccuring issue for updates since 10.14.4 as forums show. Running on early-2015 Pro.