Daredevil got super powers from an accident where he was exposed to radioactive chemicals.

That last one sounds pretty fuckin' villianious. Are you sure you're not describing Hitler?

Hawkeye and Black Widow are examples of what I was looking for, thanks! They are normal people with extremely good training.

Daredevil is one I was kind of ruling out in my original post since he gets his powers from chemicals spilled on him, so it feels a bit like a "deus ex machina" plot device, like being a mutant or an alien or touching something radioactive. EDIT: on second thought, you're right that it's known that if you (unfortunately) go blind your other senses somehow help you make up for the loss of sight and so it may be believable that by chance the chemicals he was wetted with could somehow heighten his senses, so actually Daredevil is more in line with what I was looking for than I initially thought!

Poison Ivy and Mr Freeze are amazing answers! Poison Ivy is "believable" in that she's grown immune to poison because the mix of herbs she's taken "could" have an unexpected effect and Mr Freeze too is awesome in that he's a scientist trying to save his terminally ill wife with cryogenics and somehow the experiment goes wrong.

Now you're also making me think about the very first episode of the Batman animated series where a scientist researching bats is turned into one.

Your answer suggests that a source of "believable" origin stories are experiments gone wrong, if the experiments are somehow well thought and resembling actual science

Edit: updated my thoughts on Daredevil after reading another answer on para humans in this thread.

That's a great and involved origin story. Jack takes the mantle though he wants no part in it and becomes a superhero to protect and avenge his family.

It's interesting to have superheroes who are born out of revenge, like the Punisher mentioned in another comment. I guess revenge is also part of the Spiderman lore, though it isn't involved in what makes him Spiderman in the first place (and also isn't shown acting out of rage, like the Punisher).

I admit to not knowing Starman enough so I don't know if he's shown acting out of rage (like the Punisher) or of justice (like Spiderman)

That's right! In a sense, Batman is very grounded as a comic book in that several of the main characters (Batman and the Robins) all are well trained people but with no super powers, so they are believable!

There is another answer below about Poison Ivy and Mr Freeze which I also liked and commented on and which (coupled with your answer) points to how much of the lore around Batman is well constructed (that is, characters and villains have a "believable" and well thought origin story)

100 pushups! 100 squats! 10km run every day!

Ahahahah! Great answer and a great comic. Being a parody comic, no one (not even the characters of the comic book) believe he can be real since he's grown to become a god like creature simply by doing a fairly normal workout. Genus is so pissed off when he learns about the source of Saitama's power

I really like this. It has a similar (but different) feel than the answer involving Poison Ivy and Mr Freeze (which at the end I was abstracting as "believable experiments gone wrong with unexpected but somehow credible outcomes").

In the case of para humans you're saying, powers are well constructed within the limits of the world they live in and are related to something traumatic (and not necessarily reversible) which gives them abilities related to the event.

I admit ignorance in their respect and will look more into it. Thanks!

Wolverine is a mutant and was born that way, but the experiments added the whole indestructable skeleton on top of that. The only reason he survived the experiments in the first place was because he's got crazy fast healing as a mutant power.

However, Deadpool might fit the bill, at least in some versions. He was given Wolverine's healing factor in the same Weapon X project. It's actually super powerful and would kill him if it wasn't in a constant deadlock with his cancer which is also constantly killing him. There was a comic where Norman Osborn attempted to assassinate him by shhoting him with a cure for cancer...

I'm trying to answer everyone and you pointed out correctly something I didn't define well in my original post: I was trying to find either "believable" powers (in the sense of being well constructed) or "believable" origin stories. They didn't need to come from radioactivity only. The reason I was excluding mutants in my original post was that they have powers since they have a different gene, but that is a very "cheap" way of creating a superhero, since no other explanation is necessary!

Wolverine (as pointed out in the first answer to your comment) is born a mutant and later on given an indestructible skeleton.

Make no mistake: the post is not about superheroes being or not being cool because of their origin story or super powers. I really like Wolverine and Sabertooth!

These are all great answers. As discussed in another answer to a comment (the one on Poison Ivy and Mr Freeze), I agree that "believable" experiments gone wrong based on actual science provide a source of "believable" super powers.

On the other hand: Ghost Rider and Spawn have really nice origin stories which also place restrictions on what they can do not to make the characters god like.

Watchmen is a great answer! All characters are normal people, save for Dr Manhattan and, even if the source itself of his super power is "sketchy" (in that it involves a radioactive event that just makes it true, without real explanation), there is a long section showing us how he tried to reassemble himself very very slowly while learning how to use his new powers which makes for a great and detailed origin story.

Doctor Strange is also a great answer since it taps into something that people have believed existing for centuries (aka magic) and Stephen Strange goes on to study it to save his own hands. It's a great origin story!

Green Lantern is also awesome in that it's a very unexpected origin story with aliens from Oa and lanterns being used to give this incredible power. I would say it's unbelievable as a power (with respect to the restrictions I was thinking about in my post), but the origin story and lore is much more unexpected than so many other superheroes

That's right! Moon Knight is a former mercenary (so believable training) with split personality. Then he also receives power from a god. This is also "believable" in that he is bestowed power after making a pact.

Of course, the Punisher also fits the bill. He has no powers (like Batman) just lots of training and motivation

Your comment nailed perfectly one of two the things I was aiming at with my post which I realized wasn't as well defined as I was hoping (the other thing being a colorful or original origin story, even if the power is unbelievable).

The Phantom is a perfect example: he's got no superpowers, but he managed to create an immortality myth around himself which makes him scary to others. That's great!