Factual error: When Alan Shepard is landing on the carrier, he is flying an A-4M. Not only did that variant not make its first flight until 1970, but it was also almost exclusively (with a few minor Navy roles) a Marine Corps aircraft. The A-4M is very easy to distinguish from earlier variants, most notable are the JFS (jet fuel starter) exhaust port for the JFS100-34 self-starter on the starboard side of the aircraft at mid-fuselage and the drag chute canister at the rear below the tailpipe.
Factual error: When Yeager and X1 break the sound barrier, the sound is wrong at Pancho's bar. Scene starts with a loud roar, then 2 sound barrier bangs, then quiet. Real sound barrier noise is the reverse - near silent up to the 2 sound barrier bangs, then the exhaust roar.