An error occurred during activation. Mac Mini

See if the following helps: If you get an error when trying to activate iMessage or FaceTime - Apple Support

If you're not running High Sierra (the footer indicates Yosemite and this is posted in Mavericks) and if your hardware supports a release newer than what you're running, make an external backup or two of all of your data, and then install or upgrade and boot from an external storage device as a test, and try that. Then upgrade the internal storage, if the test worked.

Can you sign into FaceTime?

Message activation error

Have someone else attempt to sign in using their Apple ID. If they can successfully sign in and use Messages/FaceTime, it may be your Apple ID.Someone I know had this happen and I was able to sign in usingmy Apple ID. After a call to Apple Support and then being elevated to a tier 2 support person, it turned out it was a problem with their Apple ID. It took the software engineers 3-4 days to fix the problem. When calling Apple, make sure you tell them it is an Apple ID problem so you don't get charged.

You can also try signing in on another device/computer. There have been a couple of posts where people's device/computer was blocked. You will need to contact Apple for that problem also.

There is a contact link.

Apple ID Support

Apple ID Support (2)

Apple Support by e-mail or chat Select your device, Battery, Power and Charging, then Power adapter not working to get a chat option.

Apple Support using Twitter

Or fill out and submit this form.