Using Classic Outlook on Windows for personal email, calendar, and contact management
Sorry to hear that you have trouble with this situation. Are you using an @outlook.com, @hotmail.com, @live.com, or another email provider such as Gmail or Yahoo? When you open Outlook on your computers, do you receive an error message or repeated requests to enter your password? Did both computers stop sending and receiving emails around the same time?
If you're using a Microsoft email address, open https://outlook.live.com/mail/ in a browser on either computer and check whether you can send and receive messages there. If the website works but the Outlook desktop application does not, this would help narrow the issue to the Outlook applications or their account connections.