In 1997, baseball commissioner Bud Selig universally retired Jackie Robinson's number, 42. The handful of players still wearing the number were allowed to keep it. As of the film's release, only Mariano Rivera of the New York Yankees continued to wear 42 on a daily basis. Rivera retired at the end of the 2013 season. As of 2014 (barring special requests or approval), no major league player will wear #42 again.
When Pee Wee Reese shows the threatening letter to Mr. Rickey, the door behind Pee Wee says "Private" backward because the lettering is on the other side of the window. The "R" in Private is facing the right way, when it should be backward.
When Burt Shotton introduces himself to the Dodgers as the new manager, the team is wearing their gray "road" uniforms and standing in the visiting clubhouse of the Polo Grounds, the Giants' home ballpark. He asks who the Dodgers are playing, and someone replies "The Giants," which Shotton should already know.
The first establishing shot outside Branch Rickey's office, the Mechanics Bank Building, shows an elevated subway passing by. While there was a subway line at that location, it ceased operation in 1940, and was demolished soon after.