Factual error: Ray, Sara, and Kendra are left behind in 1958. However, 10 weeks later Sara and Kendra are playing the board game "The Game of Life." The modern version, as seen, didn't come out until 1960 (previously it was called The Checkered Game of Life and doesn't resemble the modern version).
Factual error: When Ray is on the moon, he's getting the spear from the flag that was planted in the Sea of Tranquility, from the Apollo 11 mission. When we see the flag, it's still standing up. However, Buzz Aldrin saw, and reported, that the flag fell over from the rocket's exhaust as they left the moon, and no one from Apollo 12 put it back up.
Factual error: At the beginning of the episode, when Snart steals the keycard from the female soldier in 1986, she is shown to be wearing the U.S. army's Combat Infantryman Badge, a combat award for infantry soldiers. It is not possible for a woman to have that award then, due to the fact that infantry was closed to women at the time.