The first baseball game on the radio in the barber shop appears to be a game between the New York Yankees and the Oakland Athletics played on June 1st, 1971. The play-by-play mentioned a double by Joe Rudi, plus two strikeouts by Yankees starter Steve Kline and reliever Jim Hardin. The announcer can clearly be heard and is none other than Phil Rizzuto
The second game in 1937 appears to be between the New York Yankees and the Philadelphia Athletics with Bob Johnson of the As hitting a home run and with Bud Thomas being relieved by Lynn Nelson. The broadcast appears to be the Wednesday, June 30, 1937 game which the Yankees won 5-1 even though the paper in barbershop has the date of June 9th.
Harold Livingston, the writer, also wrote Star Trek - The Motion Picture starring William Shatner, who guest-starred here.
The set of the western town at the end of this episode is the set of Bonanza's Virginia City.