Cold War, America
If you don't have a clue about what was going down in America during the Cold War era, then you'll miss the whole point of Dr. Strangelove. For the full run down, check out Shmoop's handy summary where we lay out all the details. The short version is that the Cold War era was a period of extreme tension and paranoia between the Soviet Union and America that lasted from the end of WWII until the U.S.S.R. disintegrated in 1991.
While the U.S. and the U.S.S.R. never fought a "hot war," meaning they didn't go at it directly, they did have proxy wars like (Korea, Vietnam, Afghanistan). Because each country was afraid of the other getting the upper hand they both built up massive amounts of nuclear weapons. Everybody who knew about this arms race was super freaked out that one of the superpowers was going to lose it, start a nuclear war, and ultimately incinerate everybody on the planet. Everyone had the feeling that this might happen accidentally, too—one false threat, one technical error, one rogue commander, some kid hacking into the NORAD computer system and getting the launch codes—and kaboom.
And there were some pretty close calls, like the Cuban Missile crisis, when the US discovered armed Soviet missiles on Cuba. That's just 90 miles from Florida, folks. Everyone thought WWIII was about to start; ask your grandparents about the "duck and cover" that all school kids had to practice during air raid drills. In a helpful film, Bert the Turtle suggested we could protect ourselves from nuclear attack by hiding under our desks.
The War Room
A big chunk of the action takes place in the ominous and claustrophobic War Room. Here, the President, the Joint Chiefs of Staff, and a bunch of other important looking dudes all sit around a giant circular table and try to solve the crisis. Kubrick insisted the table be covered with green felt even though the film was shot in black and white, to give the actors the feeling that their characters were playing a game of poker with like the highest stakes ever. (Source)
The bunker-like War Room comes complete with what General Turgidson affectionately calls "the Big Board," which is a light-up map that shows the movements of the B-52 bombers headed to Russia. Oh, and also there's a big buffet table, which, while impressive, doesn't include the fresh fish requested by the Soviet Ambassador.
Though the White House does have a Situation Room in the basement of the West Wing, it apparently doesn't have anything to match Strangelove's War Room. It's rumored that this was a big disappointment to Reagan, who asked to see the War Room immediately after entering the White House for the first time. Yeah, we can see how that would be disappointing.
Kong's B-52 Bomber
The set designers were really stressed out about getting the bomber right. It wasn't all that easy either. The B-52 was super high-tech back when the movie was made, and the exact design was top-secret. All the designers had to go off of was a picture of the cockpit on the front of some book.
What's crazy is that the design that they came up with ended up being scarily accurate. Military guys came to check it out and apparently every detail was right. This actually freaked Kubrick out, and he sent a memo to the designers basically saying, "Hey guys, I hope you didn't do anything illegal to design the B-52; otherwise, we're gonna get an unpleasant visit from the FBI."