I can't activate iMessage or FaceTime on any Apple device. It keeps saying that it could not be activated and I should try again later. I'm on the latest version of macOS and iOS and the date and time are set automatically on both devices. I have no phone number linked to the Apple ID so it has nothing to do with a carrier, etc.
First, make sure iMessage is turned on: To check if iMessage is turned on for your device, go to Settings > Messages > iMessage.
Then, are you sure your WIFI or cellular connection is strong enough. It does involve your carrier if you're trying to message without WIFI. Also, I've had issues if I'm not logged with my Apple ID, although I haven't seen that it's actually required.
I see below that you have tried rebooting. Perhaps we should have asked which device are you trying this on? I assumed an Apple iPhone, but are you talking about your desktop or laptop computer? I was also going to suggest totally turning the iPhone off; maybe after also shutting down iCloud and trying to start over from totally off. My next step would be to do this on both devices you are trying to iMessage between. Obviously you are not trying to iMessage yourself, but getting it working between your desktop and your smartphone would be a good test environment.
With these kind of issues it can help to not only turn it off, but entirely log out of iMessage and FaceTime from the device, reboot and try to log in again from fresh.