Heroes come in all sizes. Whether or not they are endowed with super-human strengths of one kind or another, it is a character's weaknesses that define him as a hero to me. Perhaps I am reading Lord of the Rings incorrectly (I'm sure there are plenty of people on a forum like this who could set me straight), but one of the things that turned me off from the start was the "externalization" of evil. That is, evil was in the ring that had to be tossed into a volcano, or otherwise hidden away from polite society. People themselves were not weak, or responsible for "evil." If that is the case, there is no heroism in overcoming it, and no accountability, either.
Kick-Ass is a fantastic study. The main character decided to become a masked hero for the glamor of helping others. After getting stomped, he repeatedly turned away from the hero business, but was lured back in again for different reasons. In the end, the character became a true hero for someone else, and at great cost to himself.
The live action Space Battleship Yamato (2010) was a very dazzling tale of heroes. The central character repeatedly faced no-win "Kobayashi Maru" situations throughout the film and grew as a human being from each encounter. The final situation demanded the highest price, and the hero paid it to rid the world of "evil."
A hero can still be a hero even if he does not die, or otherwise "pay a price" for his actions. If the potential for loss is there, then the actions are heroic.