Pain is induced in normal human beings by signals sent to the brain using the nervous system. These signals are usually caused by some kind of harm / damage to the person (i.e. a cut), but you can manipulate the nerves to send the signals even if no harm or damage is done to the body.
So pain is simply a feeling, and there isn't anything contradictory about an indestructible or immortal being feeling pain, as long as that pain wasn't caused by some form of harm / damage to the being.
So pain is simply a feeling, and there isn't anything contradictory about an indestructible or immortal being feeling pain, as long as that pain wasn't caused by some form of harm / damage to the being.