The "Black Night" mission scene shows the launch of a Mercury/Redstone mission, which couldn't put a Mercury capsule in orbit, let alone a larger Gemini prototype.
The episode centers around the test launch of a Saturn V rocket. The episode is set in January 1964 but the first Saturn V wasn't launched until November 9, 1967.
When Kim is in the hotel gift shop, she looks at images of NASA astronauts taken years after the episode is set.
When John and Kimberly are pulled over by the police (at around 5:30 minutes), you can clearly see a 3rd person in the vehicle that John and Kimberly are travelling in.