While this can be seen in several episodes its very clear on here you can see the Mayberry Courthouse. Barney & Andy must be out of town.
Eliot Ness attended meeting in Washington DC to discuss the paper robbery with the secret service. During the meeting the agent in charge showed an example of a counterfeit fifty dollar bill that bore the facsimile signature of Ivy Baker Priest who served as secretary of the treasury from 1953 to 1971, a term that began some two decades after the events depicted in the story.
Mr. Moon buys an airplane ticket to travel to Chicago in the early 1930s. People always took a train then not by air travel.