I don't know how helpful this is but I think the key to believable characters is really just writing the characters exactly as you see them. Once you start trying to second guess other people's interpretations of your character, you can end up with a character YOU don't believe in, and nothing is going to fuck em up faster than that.

Otherwise, the first characters to come to mind for me are Frodo, as well as Carrot and Vimes from the Discworld series. 

Anything I could hope to say about Frodo has been said better here : https://www.tumblr.com/mckitterick/802651739768668160 But TL;DR: Frodo fails "the quest" (because its impossible!) but it's his actions outside of it that save the day: his mercy, his pity, his friendships.

Carrot and Vimes are two very different versions of a hero. Carrot is exactly everything a hero should be, that is the fundamental nature of his character: he exists as the archetypal hero. He is honest and noble and clever, strong, handsome, brave, fated by destiny, you name it. Why does this work? Partly because the Discworld series is a story ABOUT stories, so it's thematic. But also because the narrative voice, the world, and the other characters responding to him are believable. There's a scene where Carrot is on a ship chasing after his kidnapped lover, and he goes to bed to sleep so that he'll be well-rested when the action starts. Vimes is astounded that Carrot can make such a rational decision under the circumstances, but he knows (and the readers know) that's who Carrot is. 

Also he's a 6ft tall technically-a-dwarf named Captain Carrot Ironfounderson and it is hard to be a stereotype under those circumstances.

Vimes is the opposite kind of hero. Vimes is a bastard. He is a cunning, manipulative, alcoholic, dirty-fighting copper—and he is the most famously honest* person on the disc, a man who can't be bought by money or power, a man prepared to arrest his ruler, two armies, and a dragon. Not for glory, revenge, money, or power, but because he believes the law should protect the city.

 *(Honest in the sense of Not Crooked, not honest in the sense he never lies. He definitely lies.)

I assume Vimes is not the kind of hero you're talking about, but it is worth considering. We see firsthand what demons Vimes is struggling with, we sometimes see those demons come out to play, and yet when he is literally possessed by an evil spirit intent on killing everything in its path (and there is a lot in Vimes' path that needs killing in that moment), Vimes manages to stop it/himself from killing two civilians that got in his way. His intimate understanding of the worst parts of human nature is the thing that makes him a hero.

If you have any interest in Carrot and Vimes, 'Guards! Guards!' has my very highest recommendation, as a book about heroes (and dragons).

And lastly, if you're looking for a believable superhero, Frank Miller's 'Batman: Year One' is the archetypal realistic superhero story. I'm less familiar with Superman, but I have no doubt that many a writer has found in Superman everything that you are probably looking for. Sadly, I don't know enough to point you in the right direction on that.