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).
Character mistake: After Ray Palmer says he's an Eagle Scout, he says to Leonard Snart, "be helpful to others, scout's motto." The Boy Scout's motto (including Eagle Scouts) is "Be Prepared." "To help others at all times" is part of the Scout's oath, but that's different than its motto and not something an Eagle Scout would mix up.
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.
Character mistake: When the Legends are talking about Chicago in the 20's, during prohibition, Nate says there was "illegal drinking." But prohibition didn't make drinking alcohol illegal. It was only the production, importation, transportation, or sale of alcohol that was illegal. As a historian, Nate would know this and not make that statement.
Continuity mistake: In the scene aboard the Waverider when the crew discusses their discovery of the Oculus, the first two times you see Sara sitting on the Captain's chair from the back she's leaning forward with her elbows on her knees. When you see her from the front, though, she's sitting with her legs crossed and her right hand on the control panel, and in a following shot you can see her actually change her position and lean forward.
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.