It's Good to Be Back on the Moon - S1-E9
Factual error: The Apollo 11 flag gets knocked over by the Chinese rover...but the flag was blown over by the Eagle's takeoff thrust, witnessed by Buzz Aldrin. This was confirmed years later by the Lunar Reconnaissance Orbiter, which showed shadows of the flags planted at all six Apollo landing sites, bar Apollo 11's. The other 5 were all planted further from their landing craft precisely for that reason. It's also widely assumed that the flags on the moon would have been bleached white by decades of unfiltered sunlight, not in good condition like the one shown.
Factual error: Emily is in Paris when her boyfriend calls from Chicago and wakes her. She asks the time which he says is 7pm, and she says 3am in Paris, which cannot be. Central European Time is 6 hours ahead of NY and 7 from Chicago. 7 pm in Chicago would be 2 AM in Paris.
Factual error: At the end of the episode, as the veterinarians are leaving the pub, the camera pans up to the sky and you can see the full moon right next to the north star and the big dipper. In the northern hemisphere the moon is always to the south.