No party ever created any Doomsday Device of the sort talked about in the film — a computerized giant bomb meant to create radioactive contamination on a global scale. The Doomsday Device was meant to be a metaphor for the nuclear arms race in general, not an actual weapon. It is based on legitimate concerns about nuclear contamination (stemming from Leo Szilard's initial idea of a "Cobalt bomb" in 1950), and fears of automation of command and control systems. Both the US and Soviets did look into the question of full-automation of such systems, both backed down from the idea because of (legitimate) fears of machine error. In retrospect these fears are pretty well-supported — both sides had computerized systems that did create numerous false-alarms, detecting nuclear attacks that weren't happening.

Both had early-warning and detection systems that did, however, allow for various combinations of machine and human judgment. The Soviet "dead hand" system, for example, which came some time after the Strangelove film, did involve a quasi-automatic means of triggering a nuclear attack under certain circumstances. But even this required human intervention. The US did, in the mid-1950s, look into the possibility of massive bombs that could be buried and whose main output would be contamination. They never built them.

The question of nuclear weapons being used without express authorization was a real one in the early 1960s. The US had created tens of thousands of warheads at that point, and the controls and locks on them were essentially nonexistent. The idea that a general could accidentally start World War III was a real concern of the Kennedy administration, and in the early 1960s he mandated that all nuclear weapons in foreign territory (and eventually all domestic nukes as well) have security in place to prevent unauthorized use. The Eisenhower administration had qualms about this as well, though Eisenhower did allow certain officers to initiate nuclear war under specific circumstances (this is known as "pre-delegation" of nuclear weapons) — the fear of a Soviet attack was so great that a great many nukes stationed overseas could have been used by basically anyone who had physical access to them.

The Soviet ambassador's behavior is modeled on Andrei Gromyko, the actual Soviet foreign minister for several decades. It is an outsized version of him, of course. But Soviet behavior in the UN from the 1940s through the 1960s was one of general opposition to the US, general mistrust, and so on. Buck Turgidson is modeled off of a Curtis LeMay type, who did advocate preemptive war and "winnable" nuclear attacks. Dr. Strangelove himself is an amalgamation of several Cold War scientist figures, notably Wernher von Braun (ex-Nazi missile designer), Edward Teller (Hungarian nuclear scientist with a fetish for big bombs), and Herman Kahn (RAND Corporation strategist of deterrence and escalation).

Anyway. The general gist of the movie, that accidental nuclear war was possible, that both people and machines could fail catastrophically, that the essence of nuclear strategy was one of paradoxes and contradictions (the greater the danger, the greater the safety, etc.), that generals and think-tank scientists could be coldly murderous in their implications, and so on, is not so far from the truth in exaggerated form. It is not quite as exaggerated as one might sometimes believe, though — the truth is sometimes quite incredible itself.

Eric Schlosser, whose book Command and Control is an excellent and gripping history of nuclear weapons and the threats of accidental or unauthorized detonation, has written a nice piece on the New Yorker's website about the historical validities of the movie. The movie is, of course, meant to be a farce, but it is a very insightful one, despite its absurdities.