One possible way for Hyronalyn to work is to inhibit those cellular processes in which radiation might create harmful mutations; essentially, it puts the cells to sleep so that even if there is damage, it doesn't get propagated since the cells aren't dividing. There would then be a limit to how long you can keep the cells from doing their work until the cure becomes worse than the disease.
On the other hand, or in addition, Hyronalyn might perform cellular culling and/or repair, at a rate that can compensate for certain levels of radiation exposure but not for others.
But we know that at least 24th century medicine, and perhaps also its Kirk-era equivalent, can utterly defeat radiation damage simply by rewriting the entire genome of the patient. We see medical miracles like this in "Identity Crisis" and "Genesis", but also in subtler cases. It seems this is chiefly done by something you can inject with a hypospray. Wouldn't seem out of the question at all that Starfleet employees could indeed be radiation-proofed by additives in their breathing air or food or drink. Since "Genesis" basically involves a cure like this running amok, there may be limitations to how much curing the additives are allowed
to do before they become too dangerous...