A recent update has caused Outlook users trouble with connecting to their Gmail accounts.  This can sometimes be solved by removing Gmail from Outlook, closing and re-opening Outlook, then adding the account back in.

When this fails, a Windows update may restore functionality.  Microsoft is "working on a patch" as of Wednesday, March 18.

If both these fail, here are some workarounds:

Unable to add a Gmail account
Users get errors syncing Gmail and Yahoo

Reminder: you can always access your email by logging in directly to Google: https://mail.google.com/