When Roger Maris is talking with his wife from a payphone after the birth of their son, Randy J. Maris, he's talking on a payphone that is a single-slot model. In 1961, the three slot version (25 cent, 10 cent 5 cent) was still in use. The single-slot phone was not introduced until 1965.
In the bar scene, when the other Yankee players were talking to Roger Maris about Mickey Mantle's relationship with Joe DiMaggio, Moose Skowron mentions that Joe didn't talk to him at all his rookie year, until the World Series. Skowron did not join the Yankees until 1954, DiMaggio retired after the 1951 season, therefore, the two were never teammates. Plus, the Yankees did not even make the World Series in 1954. Even though the team won 103 games (the most for a Casey Stengel-managed team), they finished a distant second to the Cleveland Indians who won a then-American League record 111 games.
The broadcast of the opening day game describes the Twins pitcher as "Pascual" - a reference to longtime pitcher Camilo Pascual. He did not pitch that day - Pedro Ramos pitched a complete game for the Twins.
When Mickey Mantle and Roger Maris are playing catch, a reporter tells Maris that Babe Ruth hit .343 the year he hit 60 home runs in 1927. Ruth actually hit .356 that year, and never hit .343 during any of his 22 seasons.
After the commissioner's decision to make separate records for the 162-game season, Roger Maris is answering questions while playing catch. When he stops to talk to the reporter (clearly breaking from playing catch), he had just thrown the ball to his partner. When he's done talking, he immediately throws the ball again to his partner, though he never received it during his break.
A reference is made to President John F. Kennedy interrupting a news conference to mention that Roger Maris has hit #47 and #48. Kennedy did not hold a news conference that week in August, nor ever mention Maris during one. President Kennedy would never interrupt a press conference, let alone interrupt one to praise the accomplishments of Yankees' player. He was too big a fan of the Boston Red Sox for that.
In the game in Baltimore, the movie shows Hoyt Wilhelm coming into the 9th inning in mid-inning prior to Roger Maris attempting to hit his 60th home run. In actuality, Wilhelm started and completed the entire 9th inning; Maris was the 3rd and final batter of the inning.
The third deck at the stadium was added digitally in post-production. When Mickey Mantle and Roger Maris are doing batting practice near the beginning of the movie, in two shots you see the ball go into the stands, and fall through the third deck onto the second.