"He lost his fingers in a machine many years ago."
Translation:Han mistede sine fingre i en maskine for mange år siden.
Hans would mean some other boy/man's fingers. In this context, since they are his own fingers, only "sine" is correct. For the third person singular, the possessive pronoun is sin/sit/sine if the object belongs to the subject. For example if you say "Han elsker sin søster" you are saying that he loves his own sister. If you say "Han elsker hans søster" you are saying he loves some other man's sister.