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.
Suggested correction: I just watched the episode and the tire tracks look to be the same shape as the tires from the wagons that are moving people around (such as the firefighters). It didn't look really thick like automobile tracks, so those seem proper to appear.
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.
Suggested correction: Charles was riding with the firefighters. It wasn't Charles that took the horse - it was Patrick (Mary's friend) who took it. However, yes, there easily appeared a stunt double who looked much older and heavier than Patrick trying to actually pull the balloon down (who easily could have been mistaken for Charles).
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.
Suggested correction: The stuntman doubled for Patrick (not Charles). Charles was seen the whole time as being in the wagon with the firefighters. Patrick stole the horse and went off on his own to save Carrie.
Factual error: When the man is holding onto the rope from the balloon, you can see modern tire tracks in the field.
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.