In the Voyager episode One the crew is put into stasis chambers when they attempt to enter a nebula and the entire crew suddenly erupts with painful blistering wounds.
The doctor says that
even the briefest exposure destroys organic tissue.
If the radiation was able to penetrate Voyager's shielding and hull, how does putting the crew in statis prevent their organic tissue from taking damage?
Stasis limits bodily functions and slows the aging process. The individuals are placed in a stasis unit and can be suspended for up to hundreds of years while being less susceptible to environmental hazards.
While it wasn't explicitly stated the process by which that particular radiation in that episode destroys tissue, we can presume that, like other forms of radiation, it creates errors in replication of cells. By the events of that episode, it likely accelerates cell growth as well, creating out of control lesions and tissue damage.
If that's not enough hand-waving, perhaps the stasis units themselves have additional shielding that it is simply not economical in terms of energy costs to use to cocoon the entire ship. By placing everyone in stasis units, they could each be shielded in suspended animation, asleep, without the need for eating, bodily functions and the like for the duration of the trip.
Within a stasis field, time slows or stops for the affected items. It is apparent that the affected area is able to be penetrated with light in the visible spectrum (and so should be able to be penetrated by other EM radiation), but physical force doesn't work, so it probably can stop particulate radiation forms. It is also apparent that the entire stasis affected area is able to be moved by outside force.
Memory Beta (the non-canon Wiki) has an entry noting that a DS9 novel used stasis fields, as well.
Stasis fields of various kinds also appear in the Animated Series; in More Trouble, More Tribbles, the Klingon one has an effect solely upon ship systems; the crew remain able to function and to direct robot freighters outside the field. BEM has a similar effect via a non-corporeal being.
In The Slaver Weapon, we get the following:
SPOCK: These stasis boxes are the most remarkable thing the Slavers ever produced. Time stands still inside a stasis box. A billion years means nothing in there.
There are stasis field references in TNG:
The Child,
Unnatural Selection,
Clues,
The Host,
Man of the People,
Perfect Mate,
True-Q,
Face of the Enemy,
Starship Mine,
TNG Movies: Insurrection
There are references in DS9: The Passenger, Vortex, Life Support, The Die Is Cast, The Assignment, Empok Nor,
There are references in Enterprise: Home (as a kitchen appliance), Cold Station 12 & The Augments (as lab equipment)
Other Voyager references: Innocence, Unity, The Gift, Equinox (Alien ones that are multi-purpose), Dragon's Teeth, Blink of an Eye, Collective, Repression
There are other referenes as well; I'm not going through every transcript. Suffice it to say, however, that various forms of stasis generators of various efficiencies and specific effects exist in Trek Canon, and not all are the same effectiveness at time slowing, not all affect all material inside, and the Slaver one in TAS is about the most efficient one.
Well, in the episode, the stasis units were described to have independent life-support systems. This might indicate some kind of automated technology that keeps track of the body's functions in real time at all times and continuously regenerates it, or has 'inoculations' that the body is exposed to that act as a rapid regeneration ability should the need arise.
Plus, each stasis unit probably has extra shielding in place (they might be made from Osmium alloy from which the NX-01 catwalks were made of - that material is surprisingly resistant to radiation it would seem).
