The line up of Corvettes driven by the Apollo astronauts is accurate. A Cadillac-Corvette dealership offered special terms to astronauts throughout the 1960s and most owned or leased them. Some even raced them on NASA runways.
During his speech Gene Kranz says that "Failure is not an option." This is the title of his autobiography in real life. Kranz never said this; Ed Harris (who played him in Apollo 13) ad-libbed it during filming and Kranz loved it so much he took it for the title of his book.
In fact, Alexei Leonov trained to fly the Soviet lunar lander and was selected to be the first Russian to land on the Moon. However, the unmanned Soviet N-1 "Moon" rocket failed all four times it was launched and the Soviet Moon landing program was canceled.
At the bar during the Apollo 11 mission, Gordon Stevens talks about an event in Chicago the previous week, and describes watching the police beating kids. He is talking about the Days of Rage demonstrations by the Student for a Democratic Society association. The protests occurred from 8-11 October and were part of the movements against the Vietnam War, where 800 students faced around 2000 policemen and members of the National Guard, resulting in more than 35 injured and over 250 arrests.
Inside the command module, Michael Collins is shown to have grown a mustache. This is historically accurate, and he kept his famous 'moonstache' through quarantine after returning to earth.