Can't activate iMessage or FaceTime on any device

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.