It is one of the best Cold War/nuclear doomsday movies made for its era. In my opinion, the film does not get the recognition it should receive because it and Dr. Strangelove were released around the same time. Dr. Strangelove is a classic but was a combination of satire and humor about US/USSR confrontation. Most audiences will remember it and forget other contemporaries.
On the other hand, Fail Safe was strictly a dramatic, deadly serious suspense thriller. You could cut the pent up tension in the movie with a knife. It is from the late 1950s/early 1960's era of movie-making but I bet young folks would be surprised how good it actually is despite its extremely dated look.