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

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

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.