So, first of all that door isn't thick enough to stop gamma radiation, all gamma radiation would get through. Which is fine depending on how close you are to the blast radius; if you were very close then a lot of radiation could just come through the door, potentially ghoulifying you? (if we go lore friendly). If you were further away from the blast then not enough would come through to significantly damage you.

The real danger is if the door isn't properly sealed, which it most likely isn't judging by how easily the player can open it. Gamma radiation is only dangerous in very large amounts, beta radiation (which could leak through) can cause severe burns which would most likely not kill you but would hurt. One thing that could kill quickly (and very painfully) would be if your food gets radioactively contaminated. If alpha particles get in your body your pretty much dead as they penetrate your organs causing failure.

So yeah, you wouldn't have a good chance of survival in one of those death chambers, even if you didn't die from the severe amount of gamma radiation, you'd still get cancer or radiation poisoning at some point. Still, it could be better than going through a Vault-Tec experiment.