It's 1962, communist Cuba had gone nuclear, John F. Kennedy had the entire island under quarantine, Nikita Khrushchev was not intimidated by the young president and Kevin Costner's reputation as a legitimate actor was on the line.
For real, they cloned Bobby Kennedy for this movie.
In the center of this hot-zone was the nuclear-armed Soviet Foxtrot class submarine B-59, which on October 27, 1962 decided whether you personally would be alive right now. While surrounded by a group of 11 U.S. destroyers and the aircraft carrier USS Randolph, the submarine was eventually subjected to a barrage of depth charges.
Taking this as the opening shots of WWIII (which they kind of were), Captain Valentin Grigorievitch Savitsky ordered the B-59's nuclear-tipped missile be launched in retaliation to the U.S. surface ships. Had this been the case, it is likely that the U.S., USSR, Cuba and most of Europe would have had a full shooting-war on their hands, cowboy hats and all.