Large swathes of Apollo 13 do that for me. But oddly, the one that stands out most in my mind is a scene where Tom Hanks's character is taking his wife out to dinner. The point of the scene is a conversation in the car, but there's a throwaway reference to Pe-Te's (the restaurant they're going to).

I believe Pe-Te's has gone out of business in the past few years, but back in the early days of the Johnson Space Center there wasn't much of *anything* in the vicinity. Pe-Te's was one of the few eateries out there in the Texas coastal bayou/swamp, and it was sort of "adopted" as a semi-official Astronaut Hangout.

For some reason, that's more meaningful to me than all the places where they get the tech details right.