Factual error: When the hot air balloon comes down with Carrie in it you can see N410AA printed on it. This is commonly called a tail number on planes and is issued through the FAA, which of course did not exist during the time period of the episode.
Revealing mistake: During the scenes when they are chasing the balloon in the prairie, the tire tracks from the camera car are easily visible.
Other mistake: When Charles takes a horse to chase after Carrie in the hot air balloon, they use a stunt double. He's not even dressed like Charles. Then a few minutes later, they use the exact same stunt man to jump on the hot air balloon. Just looks silly.
Revealing mistake: When Charles steals a horse to chase down the balloon with Carrie in it, you can see it's a stunt double. Charles is wearing his hat, beige shirt, and his knee-high boots with pants tucked in. The stuntman has a pink shirt, no hat, and long pants with shoes. Maybe even white socks.
Factual error: When the man is holding onto the rope from the balloon, you can see modern tire tracks in the field.
The Rivals - S4-E14
Continuity mistake: Laura almost bumps into the boy after they get out of the lake and she looks as if she wants to kiss him. In his point of view, he is standing further away from her. In the next shot she is a few inches away from him.
Deliberate mistake: When Mary gets to the school where she is going to be teaching, all the children are standing outside wearing shoes. However when they come in to the school they are barefoot. I have a feeling this was done to show how poor the community she was teaching in was supposed to be, but it looked like too cold and rainy of a day to let the actors be outside barefoot.
I Remember, I Remember - S4-E16
Continuity mistake: Charles is delayed getting home on his wedding anniversary because a wheel on his wagon broke, he had no jack and he had to make repairs during a heavy rainstorm. Rain is still falling late at night and can be seen through the window when Caroline hears Charles arriving. She runs outside to greet him and suddenly there is no rain.
Factual error: A modern day fire extinguisher is shown on a beam Charles walks next to in the town of Bixby.
Suggested correction: Wasn't Mary the one in the balloon?
Mary was in the balloon earlier in the episode with the man who runs it but near the end, Carrie wondered off into it and it took off with her inside ALONE.