Continuity mistake: In the scene where Billy is having ice cream with his daughter, watch the can of whipped cream change even though it wasn't picked up.
Revealing mistake: In the scene of the 20th game (the following shot after Brad Pitt leaves the field) we see Jonah Hill looking at him. An extra to the left pulls out his smartphone (which in 2002 was years away from release) and proceeds to take a picture of Jonah Hill.
Factual error: The NBA arena adjacent to the Oakland ballpark has the Oracle label on the exterior. At the time, the venue was simply known as The Arena in Oakland. Oracle purchased the naming rights in 2006.
Factual error: When Billy Beane is listening to Game 5 of the 2002 ALDS on his portable radio, we should either be hearing the Athletics radio announcers or ESPN Radio's coverage. Instead, we keep hearing the voice of Thom Brennaman, who was calling the game on television for the Fox network.
Factual error: When Beane takes his daughter to look at guitars, she plays and sings Lenka's "The Show", a single which wasn't released until 2009, despite the movie taking place in 2002-2003.
Continuity mistake: In the scene where Billy is trading away players with Pete watching, his coffee cup moves back and forth from right to left without him touching it.
Factual error: Scott Hatteberg shouldn't have had to realize he could increase his chances of getting on base by taking more pitches and getting walks; in real life, he was always willing to take called strikes instead of swinging at pitches he knew he couldn't make contact with. That ability to take a lot of pitches and get on base was the primary reason Billy Beane was so eager to sign him in the first place.
Factual error: Boston Redsox stadium did not look like that when Billy visited it in the movie. Fenway Park has gone through multiple renovations since then. The most obvious errors included the concessions section he enters Fenway through was newly renovated since '02, the first shot of Fenway includes the '04 & '07 banners, and a later shot included the wall with the monster seats, which did not yet exist.
Factual error: In the scene where Billy (Brad Pitt) comes to talk to Scott Hatteberg about joining the A's, Scott wears Nike+ shoes, although Nike+Ipod was only introduced in 2006, whereas the film takes place around 2003.