Clearly, we have to look to the Cold War for an answer to this question. I would argue that the closest the world ever came to a conflict between the United States and the Soviet Union, which would have engulfed much of the rest of the world as well, was the Cuban Missile Crisis of 1962. The incident was far too complex to outline here, but there were a number of flashpoints in the crisis that could have resulted in nuclear war. Had Soviet ships attempted a sustained challenge to the blockade around Cuba, or American planes attempted to knock out anti-aircraft positions in the country, a course of action which was favored by many of Kennedy's advisors, war would have been unavoidable. American leaders learned well after the crisis that the Soviets had placed tactical nuclear weapons inside Cuba as well, weapons which would have been used against an American invasion force. So this was probably the moment when the world came closest to World War III.
The Cold War, and more specifically, the Cuban Missile Crisis.
I agree that it was the Cold War.