On the Beach fails the realism test in two ways: scientifically and behaviorally. The first is forgivable; 40+ years ago, there was less understanding of what nuclear war would do the planet. But the second aspect -- human behavior -- is where the story fails. The idea that people would get up, wash and shave and dress, go to their jobs, peacefully obtain their rations of food and so on, with a cloud of certain death getting closer every day... it just doesn't ring true. Looting, pillaging, murder, and general anarchy seem much more likely.
Nevertheless, the movie -- while very melancholy (or depressing, as many reviewers have said), is worth watching. (Especially, as some have noted, for Astaire's performance).