That was Delta Radiation, for which, interestingly, MA says the following, (note highlighted word)
Delta radiation was a highly energetic release of electromagnetic particles which can be extremely dangerous with prolonged exposure. Delta radiation could be created by the warp cores of Federation starships and some natural astronomical phenomena. Exposure to intense delta radiation could lead to cellular disruption. (ENT: "Daedalus")
"Could" be produced. Would not seem to be part of standard operation, but a possible result of an anomaly. Radiation from the warp core meant something was seriously wrong.
But "delta radiation" is imaginary. There is no such thing. It's just a bit of techobabble. In real life, ionizing radiation tends to be categorized as alpha (high-energy helium nuclei), beta (high-energy electrons or positrons), and gamma (high-energy photons), so Trek writers who wanted to sound futuristic would make up additional, imaginary Greek-letter categories like delta and epsilon and theta radiation. So delta radiation was obviously not meant to be the only kind there was, but an additional category beyond the ones that were already known today.
And
every matter-antimatter reaction produces gamma radiation, as I said. That is not optional. That is just what happens. That is E=mc^2. Every particle and its antiparticle has a known mass, and when they react, they produce a known amount of energy, which corresponds to a photon of a known wavelength in the range we call gamma rays.
IIRC "The Catwalk", where the crew have to take shelter inside the nacelles, might very well fit your bill. Although maybe that's worse, given in reality they'd be impossibly polluted to purge and then walk around unprotected.
Well, radiation is not pollution. People have a tendency to treat radioactive material and radiation as if they were the same thing, but that's like treating a guitar and the sound it makes as the same thing. Being exposed to radiation is like having a light shine on you. Once the light source shuts off, the light is gone. It's different from actually being contaminated by radioactive material like plutonium or nuclear fallout. That's actually having a physical substance in the environment that emits radiation. That wouldn't happen with a warp reaction. The end products of a deuterium-antideuterium annihilation reaction (leaving out the decay steps in between) would be gamma-ray photons (basically just really intense light), neutrinos (which pass through everything and have no effect), and electrons or positrons. So it's just radiation, no material to speak of. There is no such thing as "antimatter waste," despite the idiocy of
Voyager's Malon episodes.
However, it is possible for many substances that are exposed to sufficiently intense radiation to have some of their own atoms transmuted into unstable, radioactive isotopes, which would then give off radiation of their own. So realistically, it probably wouldn't be safe inside the nacelle catwalk even with the engines shut down. And it would probably take days after shutdown for the metal and the atmosphere in the catwalk to cool to survivable temperatures. After all, contrary to popular myth about the instant-freezing effect of space, vacuum is actually an excellent insulator, so the nacelles would cool down rather slowly.
The design of the warp core never really bothered me. I figure, if they have the technology to harness the power of matter/anti-matter annihilation into a feasible FTL propulsion system, then they were able to engineer some sort of advanced, compact shielding for the warp core components.
Sure, it's common enough in sci-fi to posit some super-advanced sciencey solution to a problem that can be solved in a simpler way, like force fields instead of walls or tractor beams instead of cables. But the problem with that is that the first rule of good engineering is to keep it simple. If a basic, no-frills solution like a wall works, then you use a wall instead of wasting energy on something complicated and unreliable like a force field. And if you want to shield yourself from radiation, the most basic, straightforward, infallible way of doing that is with
distance. The greater the amount of either space or solid matter between you and the radiation source, the less radiation can reach you. That's using physics -- the inverse-square law -- to your advantage. If a radiation-blocking force field fails, you're cooked. But having the reactor just be a safe distance from the inhabited parts of the ship is a lot more foolproof. The distance isn't going to go away if the power fails. So just because a solution is more high-tech, that doesn't make it smarter.