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.
Factual error: Laura Ingalls Wilder was born in 1867. Ralph Waldo Emerson died in 1882, which means she was at the maximum 15-16 years old (and already married) in order to meet him IF he were to appear in 1882 which is not possible because: The problems with his memory had become embarrassing to Emerson, and he ceased his public appearances by 1879 (according to wikipedia).
Factual error: This episode was mentioned by Laura at the beginning as taking place in December 1889. Rose was born in December 1886, so she should be 3 years old now, but she is shown through the episode appearing only to be roughly 6 to 12 months old and still in a crib.
Factual error: Jason mentions to Laura that someday there will be a "steam powered airplane." The word "airplane" did not even exist at that time - "flying machine" or so would likely be used during that era.