The correct answer is, the two words in the excerpt of Homer’s Odyssey that have a meaning similar to “punishing someone who that hurt you” are the following.
The first one would be “How vain the efforts to avenge the wrong!” Avenge would be the word.
The second one would be “Is the foul fiend of the human vengeance freed.” Vengeance would be the word.
Avenge is a synonym of retribution, to pay for the consequences of your acts. As well as “Vengeance”, to inflict pain or punishment for the damage caused by the actions. That is why the two words in the excerpt of Homer’s Odyssey that have a meaning similar to “punishing someone who hurt you” are “avenge” and “vengeance”.