During the opening salvos of the second Cylon war, the Battlestar Galatica was engaged by several Cylon basestars. During the engagement the Cylons launched an atomic weapon which struck the Galatica and detonated. My question is this: How could any ship survive being struck by a blast of an atomic weapon and not either be totally destroyed, or have its crew irradiated by the radiation given off by such a blast?
Nuclear explosion damage comes in three flavours:
Prompt (mostly gamma) radiation
Type 2 is only applicable in an atmosphere. There is no blast in space as there's no air to create the blast.
Type 3 is expected to be sustainable by a space worthy ship. Interstellar space is a very radiation rich environment, so we can assume the ship is radiation proof.
This leaves direct heating. In space this would consist mainly of radiative heat transfer. Large amounts of this would be directed away from the ship into empty space. A large amount would still be absorbed by the ship itself and would no doubt cause substantial damage to the ship, but not necessarily enough to destroy it by vapourising it. Most metal objects survived in Hiroshima for example. Given the advanced tech here, they no doubt have equally advanced materials technology resistant to this sort of damage (as we indeed see throughout the series).
Probably the greatest threat to the ship would be via electro-magnetic disruption of its sensors and communications, disruption of flight ops, and destruction of unarmoured critical weapons systems exposed on the ship's surface.
Something important: the Galactica is big - in the 1.5 km range, if I remember correctly. That would allow for several meters of armor on its hull. Even with the materials in our disposal, that would make a rather tough nut to crack, even with nuclear weaponry. And that would be even harder considering the materials probably available in the BSG universe...
I won't try to repeat the excellent answer of WOPR.
It is also not stated what range of yield of weapons used in Battlestar Galactica. Yields of weapons tested by the USA and USSR/Russia have ranged from the equivalent to 10 tons of TNT at the low end to 50 megatons of TNT. The largest weapons have tended to be air-dropped weapons rather than missiles due to the constraints placed upon the design of needing to be propelled.
A Trident II missile can carry a little under 6 megatons in total (12 x 500kt warheads) as a concrete example of a missile deployed weapon.
Perhaps to be effective as a weapon in space with moving targets a missile has to move relatively fast - leading to smaller warheads.
Of course any direct hit with a nuclear weapon will vaporize any armor Galactica can have (if normal physics apply) - we are talking about a ball as hot as the center of the sun. See here for more about nuclear weapons than you ever wanted to know. And here is an online calculator for the effects of nuclear weapons on armor. Just try a range of 1 meter and check the armor thickness.
On top of the points made about the lessening of the effectiveness of a nuke in space, combined with not knowing what yield the missile carried - likely to be smaller, as it was fired from a raider, rather than ship-launched - I think it's worth mentioning that it hit one of the hangar pods (IIRC) rather than hitting the main hull.
The pod in question was shown to have major structural damage and fires inside the armoured layers; when you consider how a goodly portion of the hangar pods are empty space, I'd say that if the hit had been taken on the inner hull, Galactica would have been crippled or worse. But, taking the hit on the pod - which curves more radically than the main hull - meant that the damage was directed towards less critical areas, which presented less hull area to hit.
I think the fact Galactica was able to proceed with damage control unmolested leads us to underestimate how bad a hit she took. And let's not forget Tigh knowingly vented a bunch of people into space because the ship was very close to exploding from secondary damage.