Kids, don't subject yourself to dangerous levels of gamma radiation or juiced up arachnids that you suspect might harbor mutagenic properties. That should go without saying, of course, but comic books -- particularly Marvel's -- have worked long and hard to make people think these tragedies are actually a gateway to fantastic powers.
It's true that Bruce Banner, Peter Parker, and even the vast majority of the X-Men can trace their abilities back to radiation. There are, however, other factors at play which go beyond our puny real-world science. These "factors" go by the name of Celestials, and they're responsible for the lion's share of superpowers on Earth-616.
They're also not super nice. Celestials are your typical "unknowable" entities with origins and motives that go beyond human understanding. Think Lovecraftian Great Old Ones that look like leftover props from 1950s sci-fi films. The latter facet of their design coming from creator Jack Kirby, who loved himself some The Day The Earth Stood Still.
Besides that they're menacing, not much is known about Celestials even decades after they debuted in Eternals #1. They don't really deign to speak with mere mortals -- much like Galactus in his early encounters with Earthlings. They basically just flit from planet to planet, sowing the seeds of weird genetic experimentation wherever they go. One such Petri dish was, naturally, Earth.
The Celestials are directly responsible for at least two superpowered species in the Marvel universe. The first being the godlike Eternals (who don't really show up much in the comics these days). The second group being the Deviants, who helped sink Atlantis when the Celestials swung back to our neck of the woods to check on human evolutionary progress.
Less directly, the Celestials are to thank for Marvel's mutants as well as mutates. The latter being the in-fiction term for characters like The Hulk, Spider-Man, and Daredevil -- heroes and villains who got their abilities from unexpected exposure to mutagens, like radiation. Celestial tampering has left the basic population of Earth-616 with genes just waiting to go wild.
For some (i.e. mutants) this manifested naturally around the time our planet's background radiation spiked in WWII. Others need a greater kick -- say a gamma bomb to the face -- to join the party. Which is why the Marvel U. gets a friendly neighborhood Spider-Man instead of a cancer patient, and one more reason we're living in the worst of all possible universes.