Factual error: When they first assault the city, the Greeks construct catapults. Only problem: catapults were invented at 400 - 300 BC, i.e. about 800 years later; and the first models looked similar to giant crossbows. The one-armed onager and similar types (as seen here) were a later Roman invention.
The Distance - S2-E1
Factual error: Julie picks up Marissa's iPod and presses the menu button, and Marissa acts as though the song has stopped, when pressing the menu button wouldn't do anything to the song - all that would do is have the playlist come up on the screen.