The best answer is probably just magic, though I do think their is a half-reasonable one: radiation exposure annihlates DNA. Look at the case of Hisashi Ouchi, a Japanese nuclear worker who was exposed to an equivalent dose of radiation at the center of the Hiroshima blast. He lived for 83 days in absolute agony, and looked worse than any ghoul (if you haven't seen the pictures, and have a very strong stomach, take a look, one of the more horrific things out there.) His white blood cell count dropped to zero and his chromosomes were utterly shattered, unrecognizable and their proper order unidentifiable.
Following this, what ghouls would be is an emergent property of radiation exposure. DNA reduced to a slurry, unrecognizable. However, we can postulate that once direct exposure to radiation has ended if it hasn't killed the victim, the DNA may reformulate. That is to say, the inherent structures of DNA allows it to partially come back together, execute basic functions but divorced from proper order. Think of it like a jury rigged engine - it has a doorknob for a cylinder, it's running on vodka instead of gasoline, and half of its rotational energy is devoted towards somewhere completely pointless, but it works well enough to sustain the most basic of life functions. Another way to look at it: you have a beautiful chocolate rabbit that you've melted and let settle out into a mostly formless lump.
For a better articulated example of what I'm talking about, see
https://www.angelfire.com/ego/g_saga/kaijubiologyarticle.html. Some of the underlying mechanisms are different (and make this concept more sensible than my ghouls) but the basic principle is similar.
Of course, aside from being mostly handwave dressed up in jargon, it doesn't answer several important questions. For one, even if they've survived initial exposure and their DNA has partially reformulated, any further exposure to radiation should damage their DNA, if not their flesh. We are shown just the contrary, with ghouls able to easily survive highly irradiated areas. What's more, starting with the (in my opinion very cool) ability of Glowing Ones in Fallout 3 to revive fallen comrades, built upon in Fallout New Vegas with the Marked Men who's flesh is regenerated by radiation (both in lore and gameplay where Marked Men are strategically placed in irradiated areas to make them tougher), and reaching its sickening apotheosis in Fallout 4 with ghouls able to be sustained upon radiation and nothing else (not even water!), ghouls are actively helped by radiation rather than just surviving it.
Perhaps for the resistance, we can speculate that the loose amorphous structure of ghoulish DNA is less susceptible to the effects of radation as opposed to the rigid, easily shattered double helix. Even taking this weak explanation, their flesh itself should still be susceptible to the effects of radiation. And there is absolutely no explanation I can muster for radiation's healing (or even sustentive!) abilities.
The second problem is why do some people become ghouls and others do not? In Fallout 1 of course, every single ghoul we meet seems to be from Necropolis. The same is true in Fallout 2, every ghoul that has an origin story is from Necropolis. The question than becomes why did everyone in the Bakersfield Vault go ghouly, and why didn't anyone else in California? The "DNA Slurry" explanation utterly fails to explain this, though most ghoulification theories also fail and have to resort to handwaving some secret vault experiment and handwaving the lack of other ghouls as simply not depicted or elaborated upon but existing.