Cannot sign into facetime or imessage on macos 10.8.5

Not sure, have you tried signing out of iCloud & any other Apple thing you're signed into, then try signing in again?

If you still want to work on it...

To find out if it's system wide or user specific, try this...

Open System Preferences>Users & Groups, unlock the lock, click on the little plus icon, make a new admin account, log out & into the new account.

Does it work in the new account?

What you need to make and receive phone calls on your Mac

To make and receive phone calls on your Mac, make sure of the following:

• Your iPhone has iOS 8 or later installed. For more information, see the Apple Support article Update the iOS software on your iPhone, iPad, or iPod touch.
• 
  
• Your Mac and iPhone are on the same Wi-Fi network. For more information, see Connect to the Internet using Wi-Fi.
• 
  
• Your Mac and iPhone are signed in to iCloud and FaceTime using a single Apple ID. For more information, see Set up iCloud on your Mac, Sign in to or out of FaceTime, and the Apple Support article Use Continuity to connect your Mac, iPhone, iPad, iPod touch, and Apple Watch.
• 
  
• Your Mac and iPhone have Wi-Fi calls turned on.

For more information, see Make and receive phone calls on your Mac.

FaceTime requirements on Mac - Apple Support

Though I wonder why to recieve Facetime calls...

• A Mac with OS X 10.9.2 or later, and a built-in or connected microphone