Thanks for contacting Apple Support Communities!
Let's try a few things here and see if we can get FaceTime up and running on your Mac.
First let's try to force quit FaceTime.
- Go to the Apple Logo at the top left of your screen().
- Click "Force Quit...".
- Select "FaceTime", and then select "Force Quit".
- Another window will appear that has another force quit button, just click force quit on that new window and move on the the next step. If FaceTime did not appear in the force quit window, then just move on to the next steps. If you still have the force Quit Applications window open, then go ahead and close it.
- Open "System Preferences" by going back up to the Apple logo, and selecting "System Preferences".
- Sign out of your Apple ID/iCloud.
- Restart your Mac.
- After your Mac has booted back up. You will need to sign back into your Apple ID in System Preferences.
- Open FaceTime
- And from here, try to sign into your Apple ID
If you have any questions, please ask!