As to biblical inerrancy...

Jesus is mentioned as born in the reign of Herod the Great. At that time, Judea was a client kingdom of Rome. It also says that there was a census taken at his birth. The romans did not take a census in client kingdoms. They did take a census at a later date when Judea was a province, however, but that means Jesus could not have been born under Herod if the census story is true. Further, Jospeh lived in Galilee, which was not a part of the province of Judea, and so not subject to a census. A census also required the head of the household(the male) to announce his property, there would be no reason for a pregnant Mary to travel.

Thus, the story of Joseph travelling to Bethlehem to register as part of a census under Herod is obvious nonsense. That makes no difference for sane Christians who can happily appreciate the story for its moral value. It does, however, become a significant problem if you treat it as an authoritative history book.