Henry Luce: Now, I want them all to meet my people who will write their true stories, Naturally these stories will appear in Life magazine under their own bylines: For example, "by Betty Grissom", or "by Virgil I. Grissom", or...
Gus Grissom: Gus!
Luce: What was that?
Grissom: Gus. Nobody calls me by... that other name.
Luce: Gus? An astronaut named "Gus?" What's your middle name?
The entire meeting in the Pentagon after the launch of Sputnik can be considered this. LBJ is an unapologetic Large Ham and constantly misunderstands the accent of the Von Braun expy ("Pod" as "Pot", "Specimen" as "Spaceman", and "Chimp" as "Jimp"). Capped off when Those Two Guys show a reel of various daredevils and circus performers as Astronaut candidates, and President Eisenhower demands Test Pilots which causes everyone to react in horror.
Culminating in the moment when both the astronauts and their wives absolutely REFUSE to let him pose on live television with John Glenn's wife Annie, a sweet woman who suffers from a terrible stutter. LBJ, someone who rarely gets denied for anything - this is a man who bullied hard-line segregationists and got away with it - throws one epic tantrum in his limo when he's denied this photo op.
LBJ: ISN'T THERE ANYBODY WHO CAN DEAL WITH A HOUSEWIFE?!
The scene in which two officers discuss the unsuitability of Chuck Yeager for the program, unaware of the fact that he's just down the bar from them and can hear every word. Double bonus points because:
This actually happened; AND...
The bartender in that scene is played by the real Chuck Yeager.
Al Shepard's launch gets delayed, leaving him stuck in the capsule. After a while he tells Mission Control that he really needs to pee, something nobody had made any provisions for.note The flight was only going to last 15 minutes, but they didn't anticipate that the astronaut would spend a couple of hours buttoned up in the capsule waiting for launch. It goes to a full invocation of Potty Emergency, with shots of people using a water cooler, pouring tea, coffee, etc. Finally, clearly irritated, he says, "Request permission to relieve bladder!"
John Glenn follows through with the Jose Jimenez jokes and bids Al Shepard farewell with a "Vaya con Dios, José."
The scene where the seven are presented with the Mercury capsule...ah, spacecraft and the astronauts prying design concessions out of Von Braun's group, culminating with John Glenn wordlessly correcting their speech by pointing at the press (reminding them who holds the keys to the funding) every time they say something he doesn't like until they get it right.
In the same scene, Glenn trying to be diplomatic and eloquent about the way the astronauts will be perceived by the press if they aren't really piloting the spacecraft, only to be interupted by Shepard.
Al Shepard: We want a window!
When Yeager goes up for his final record-breaking flight, the control tower doesn't have any flight plans for that time, but when they realize who it is, they just assume there's a flight plan...somewhere. Be because Yeager would have one...right?
The 2020 series
When a messenger arrives to notify John Glenn that he's been selected for the astronaut program, he initially walks up to Glenn's (older, heavier, bespectacled, and generally less-handsome) coworker and asks if he's Major Glenn. The man looks back at him and dryly replies "That is the nicest thing anyone has ever said to me."
The initial pool of astronaut candidates engage in macho posturing against one another. With the exception of Deke Slayton and Gus Grissom, who become friends with just a few terse words.
Deke: Deke. Gus: Gus. [The two shake hands] Deke: Rye? Gus: Rye. Deke: You fish? Gus: Fishing's my favorite. Also cars. You? Deke: I love cars. Gus: Big outdoors. Deke: Alright, then. Gus: Yup. [They both take a drink]
The introduction to nurse Dee O'Hara, who rapidly makes it clear she won't be putting up with Shepard's bullshit when he tries to flirt with her.
The looks of pure disbelief on Kraft and Gilruth's faces when, at the moment they're finally about to get an American into space, Deke Slayton reports Shepard needs to pee.
weren't shown in the film – probably because viewers would think they were making it up.