Hank Green very briefly (in about 2 1/2 minutes) talks about whether we crazy humans could destroy the Earth several times over with nuclear weapons:
I think Hank is probably right in his conclusion that no, we probably could not (and it would still obviously be insane to test the idea).
That last sentence might lead some to leap to the erroneous conclusion that Hank Green is somehow defending nuclear weapons or even minimizing the horror of nuclear war, but of course he's saying no such thing. As he makes pretty clear, a major nuclear war would likely wipe out most humans and cause a major extinction event affecting other kinds of living things. It's nonetheless a little reassuring to think that life itself, and even a fraction of humans, would probably survive something that horrific.
Some might also fear that even mentioning this could encourage some maniac to think that a nuclear war might be winnable. It seems to me that anyone that deranged is pretty much by definition not going to be influenced much one way or the other by a rational analysis.
In passing, this reminds me of a related popular idea, namely that the "duck and over" drills of the early Cold War were crazy or naive. If you're young enough to be unfamiliar with what I'm talking about, for a while in the 1950s and early 1960s schools in the U.S. (and probably other countries) held civil defense practice drills in which students would duck under their desks and cover their heads and faces.
There is a popular belief today that such drills were a waste of time, given that a nearby strike from a hydrogen bomb meant certain death. But few civil defense officials had any delusions about that. The purpose of the drills wasn't to protect students near ground zero but those far enough away from an explosion that the immediate danger to them was from the shockwave and resulting flying glass and debris and structural collapse. In fact, duck and cover is a good practice in the event of a tornado, earthquake, or nearby conventional explosion.
An obvious response is that an all-out nuclear war would kill everybody anyway, either immediately or in a terrible lingering fashion as a result of radioactive fallout. What this overlooks is that a nuclear war might be limited. To belabor the obvious, I'm not suggesting that it would be anything less than lunacy to launch a nuclear war in the expectation that it could be kept limited. The point is that it's not impossible. For example, an accidental detonation or an attack launched by a small nuclear power (basically anything other than the Soviet Union) or by terrorists or by rogue military personnel (as in Dr Strangelove) would likely have been limited unless it provoked an all-out counterstrike and counter-counterstrike. In the event of such a limited number of nuclear explosions it would make sense to take steps to try save as many lives as possible.
The same goes for natural disasters, and in fact we have actual evidence of this. Many people were seriously injured as the result of an asteroid superbolide over the southern Urals in Russia in 2013. But as this Wikipedia entry) notes, a fourth-grade schoolteacher almost certainly saved her students from injury by having them "duck and cover":
A fourth-grade teacher in Chelyabinsk, Yulia Karbysheva, was hailed as a hero after saving 44 children from imploding window glass cuts. Despite not knowing the origin of the intense flash of light, Karbysheva thought it prudent to take precautionary measures by ordering her students to stay away from the room's windows and to perform a duck and cover maneuver. Karbysheva, who remained standing, was seriously lacerated when the blast arrived and window glass severed a tendon in one of her arms; however, none of her students, whom she ordered to hide under their desks, suffered cuts.
In those Cold War U.S. civil defense drills, students were taught to "duck and cover" on their own in the event of a bright flash, which could be evidence of a nuclear explosion as well as a superbolide. I would be not surprised if Ms Karbysheva was old enough to remember such drills from her youth and acted accordingly.