Factual error: When Mark types email addresses in to tell people about facemash, he writes to several people @harvard.edu. At the time the movie takes place, undergraduate email addresses were all of the form email@example.com. Furthermore, the network brought down by facemash would have been referred to as the FAS network. (FAS stands for Faculty of Arts and Sciences; the eponymous network covered all buildings within the College and Graduate School of Arts and Sciences.).
Factual error: The construction of Hadrian's Wall as shown at the end of the film is not historically accurate. A wooden wall was constructed first with a deep ditch on either side. One of the ditches should be visible as the characters approach the wall but it is not. The stone wall shown being constructed was not added until later and then it was constructed in sections. It did not become a continuous stone wall until some years later. In addition, the stone wall shown is not wide enough. Hadrian's Wall is around 2 meters wide at the top and even today you can comfortably walk on top of it.
Continuity mistake: Just before Secretariat's first race, the main characters are discussing tactics in the parade ring. In the background is a jockey in pink and gold colours standing by his horse. When the shot changes angles he is mounted and being led out onto the race track. This means that he would have only taken one second to mount and travel a distance of about 10 meters. A couple of shots later he suddenly appears again, but this time he is being led into the saddling-up area, even though the jockey is already on board.