Christopher Kraft (Eric Ladin) and Robert Gilruth (Patrick Fischler) tell the pilots at the Washington briefing that there will be only seven Mercury astronauts. At this stage in reality, NASA intended to select twelve astronauts. The number was cut down to six when it was realized that few or no pilots would drop out during astronaut training, then finally increased to seven.
The scene in which Glynn Lunney says he did not include any Marine pilots on the list because the Navy told him the Marine pilots were their washouts is amusing but inaccurate. In real life, there were no Marines on the initial candidate lists because Dr. Robert Voas, the Navy representative in the selection group, erroneously thought that the Marine pilots' records were in the same files as the Navy pilots, since they trained together at Patuxent River. John Glenn and another Marine were added to the list near the end of the process.
There is a subtle visual nod to Tom Wolfe's book in the Washington briefing scene. Christopher Kraft (Eric Ladin) is seen standing with his hands on his hips with the thumbs to the rear. Wolfe's book includes a passage in which pilots at a "charm school" for potential astronauts are told always to place their thumbs to the rear when standing with hands on hips. (Kraft himself never served in the military due to a burn injury he sustained to his right hand at age three.)
When going through the list of candidates, the fellow strikes names off a list with a felt marker. While some might think this an anachronism, the first mass-market broad felt-tip marker was the Magic Marker, which debuted in 1955. The actual invention itself dates back to 1910. Even so, in the early sixties, the preferred marker for NACA / NASA and the military was the wax pencil.
Both the opening reel of The Right Stuff (1983) and this episode include an Edwards test pilot's funeral attended by one of the main characters (Chuck Yeager in the film, Gordon Cooper in the episode).