You are talking about the Cold War and a doctrine of deterrence known as "mutually assured destruction." The USA and the USSR both amassed arsenals of nuclear weapons so large that they possessed enough nuclear weapons to destroy not only one another but the whole world in the process. This caused much fear and apprehension among everyday people, to be sure. But it also kept leaders from actions that would lead to a direct war between the superpowers. The Cuban Missile Crisis of 1962, for example, showed that backing away from war was wiser than creating a nuclear clash with one another.
