Hi everyone! This is my first time posting on this subreddit, it truly is a fascinating page to look at and read. For some time I've had this question on my mind: Is there any point in post World War 2 history, apart from the Cuban Missile Crisis or even the Cold War in general, where we have been dangerously close to either a nuclear war, to nuclear confrontation of some sort, or to nuclear weapons being utilised?
I look forward to hearing any responses! :)
The Able Archer 83 war scare is generally considered to be the other real-deal "closest call" scenario other than the Cuban Missile Crisis. More details here. For a book length discussion, see David Hoffman's The Dead Hand. It was a war exercise that came at a very tense time when the Soviets were genuinely afraid that it might be a ruse for a first strike attack. Under such situations it doesn't take much of an accident on the ground to potentially escalate, especially since the USA was somewhat naive about how serious the Soviets actually feared such an attack.
The other major category of "close calls" are various false-alarms that occurred in the USA and the USSR/Russian Federation. At least one of those occurred under the Yeltsin administration, post-Cold War. These generally involved early warning systems malfunctioning (including the running of war simulations without people realizing they were simulations) or mis-reading natural events (including flocks of geese and even rise of the moon!). In all cases they were kicked up to a human being who said, "hmm, probably not" and declined to do anything further.
The last "close call" category are that of nuclear weapons accidents, where if an accidental nuclear yield had been achieved there might have been enough confusion to lead to serious incident. Personally I am dubious that any of these would have lead to a nuclear war but it was a concern that various high policy makers (such as Robert McNamara) had during the tense times of the Cold War. For more on this, Eric Schlosser's Command and Control is a pretty lively account of nuclear accidents.
There was actual a very similar question posted today as well.
restricteddata's comment talks more about weapons accidents on this thread, but the top level of the other thread goes into more detail into Able Archer and includes the transition from Stalin to Khrushchev in '53.