Factual error: Before he is to ship out he has to find his "masters mates and pilots license." That is the name of a maritime union. No such license exists. He also would not be in that union as that is only for ships' officers, which he obviously is not. He would have been in the "SIU" union, for deckhands. And he would need to supply his "AB" (able bodied seaman) ticket.
Continuity mistake: Near the beginning of the film, Llewyn goes into a subway station. The train that comes in is a different car model than the one he is sitting in in the next shot. Also, the exterior shot is of a train that did not yet exist in 1961. That model was introduced around 5-6 years later.
Factual error: In the scene where Llewyn is standing on the elevated train platform, a train comes in and it's one of the trains currently in use by the NYC subway system, not a train that was in use in 1961.
Factual error: In any scene where Llewyn is using a capo on his guitar, the capo he is using is a Shubb capo. They are excellent and he probably would have used one, save for the fact they were not invented until 1980, around eighteen years after the film takes place.