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WordPress shows "Error: User registration is currently not allowed." when Anyone can register is off. Turn it on under Settings → General → Membership.
Turn on registration
In your WordPress Dashboard, go to Theme My Login → General. Under "Registration", check "Anyone can register" next to "Membership", and then click "Save Changes".

This is the same setting as "Anyone can register" under Settings → General, so you can turn it on in either place.
On a Multisite network
Multisite sites have no Membership setting under Settings → General. Registration is controlled for the whole network instead: go to Network Admin → Settings, find "Allow new registrations" under Registration Settings, and choose "User accounts may be registered".
With Theme My Login
While registration is off, Theme My Login hides the Register link on its forms and sends visitors who open the register page to this error. Once registration is on, both the link and the page work.