Hard to believe, but it's true. It's in the town bylaws, Article III, "Collecting and Dealing in Used Articles:"
2. No person shall engage in the business of buying or selling second-hand articles within the limits of the town of Amherst unless he is duly licensed by the selectmen.
Punishment is "not more than 20 dollars for each offense." This is an old, crufty law from 1939, and it's time for it to be repealed, even if it is never enforced. We should be encouraging people to re-sell and re-use used articles; it's better for the environment.
And besides, what business does the town of Amherst have in telling me what I can or can't do with my old stuff? I suppose I should go down to Town Hall and ask for a license to buy some second-hand stuff on Ebay and see what they say (prediction: I'll get the blank stare of "what the heck is this crazy person asking????").