FaceTime and iMessage - an error occurred during activation
Found this apple support article covering this exact error: - https://support.apple.com/en-gb/HT201422


  Check your device settings
  
  
  Make sure that you’re connected to a cellular data or Wi-Fi network. If you're using an iPhone, you need SMS
  messaging to activate your phone number with iMessage and FaceTime.
  Depending on your carrier, you might be charged for this SMS. 
  Go to Settings > General > Date &amp; Time and make sure that your time zone is set correctly.
  
  
  ￼
  Turn off and restart iMessage and FaceTime
  
  Go to Settings > Messages and turn off iMessage.
  
  Go to Settings > FaceTime and turn off FaceTime.
  
  Restart your device.
  
  Turn iMessage and FaceTime back on.
  
  ￼
  Still not activated after a day? It might take up to 24 hours to activate iMessage and FaceTime. If the issue continues after 24 hours,
  follow these steps:
  
  Make sure that your device has the latest version of iOS.
  
  If you're using an iPhone, contact your carrier to make sure that you
  can receive SMS messages.
  
  Contact Apple Support.


It has the links to support once you've ruled out the easy things like a setting on your Mac. They can help if your network connection isn't reaching Apple's servers or if there's an issue with your account on the server side.