Is there an NEC-compliant method of repairing minor damage to electrical wire insulation without a junction box?
You can repair anyway you want, but for it to be repaired and the UL still good, you have to use something like 3M 2234 Cable repair kit.



Here's some info on it

If it is romex you are best to cut the bad part out and use a junction box (or 2) to put in the good piece.  If it is in the wall then you might have to change out the whole run.  With ARC Fault breakers (AFCI) slowly becoming code in more than just bedrooms it could create a problem.