How close was the US to firing off a missile during the Cold War?

Very close several times. The threat of nuclear release by both the US and the USSR was very high. During the Cuban Missile Crisis, President Kennedy was ready to launch upon report of the erection of any Soviet missile in Cuba. US armed forces were put on the highest stage of alert once when the Cheyenne Mountain complex detected launches in the Soviet Union in the eighties by mistake. And there have been several other incidents, including 9/11.