You’re breathing the same recycled air for weeks at a time. The radiation is continuously creating isotopes in the gases in that air. That means a continuously increasing radiation exposure the longer you stay underwater.
On the surface any radioactive gases get dispersed in the atmosphere.
No, this is not the case. We often surfaced at night to recirculate with fresh air. In fact, this is done so often that the first thing we do is turn the clocks 12 hour forward so our day shift when everyone was working occurred at night. We need to surface to recharge the lead acid battery off the diesel engine as well, which was done to make sure that kept working well. I did this when newly qualified as an electric plant operator and it was difficult for me to do correctly.
> On the surface any radioactive gases get dispersed in the atmosphere.
The relevant radiation exposure when you're outdoors is not from radioactive gases in the air, it's from radiation hitting atoms in your body. (The relevant radiation to compare with what a nuclear reactor produces is actually not from the Sun, it's from cosmic rays.)
Cosmic rays come from the sun, as well as other sources, but mostly from the sun. Cosmic rays are highly charged particles, not a particular type or class of radiation, and are energetic enough to produce ionizing cascades in the atmosphere.
> Cosmic rays are highly charged particles, not a particular type or class of radiation
Highly charged particles are a type of radiation, at least as that term is used in the context of this discussion. (Yes, in some physics contexts "radiation" only refers to massless particles like photons, but in such a context no "radiation" to speak of comes from nuclear reactors either, since the potentially hazardous radiation from nuclear reactors is neutrons and some alphas and betas.)
> The radiation is continuously creating isotopes in the gases in that air.
What radiation? Note that he didn't say he received radiation exposure throughout the submarine, only "working on the reactor". That's an isolated space that doesn't continuously exchange air with the rest of the submarine and has an extra layer of shielding between it and the rest of the submarine (in addition to the shielding around the reactor itself that separates that from the space where people working on the reactor go--nobody ever goes inside the reactor itself).
Reactors on Navy vessels, since at least the 70s, are very safe. They are closed systems and are completely sealed in compartments surrounded by feet of shielding. While some negligible amount of radiation does get through, it's half life is very short. Radioactive isotopes don't escape, nor do they get created by proximity to a radiation source and start floating around the crew compartment. In addition, each crew member carries a dosimeter which tracks their exposure. As others pointed out, walking around outside exposes us to more radiation (from the sun, earth, ceramic tiles...) than spending months next to to one of these reactors.
On the surface any radioactive gases get dispersed in the atmosphere.