Hospitality to strangers was very important to ancient Jews. If there was no room in the inn, it was plausible to try to offer some sort of shelter, even if it wasn't ideal. Also, it was common in ancient times for women to give birth over straw (to soak up the effluvia), so it might not be so odd for a woman to give birth in a stable.

I know this is not specific to your question, but there is another part of the Nativity story that casts doubt on its veracity: The whole reason Mary and Joseph are going to Bethlehem, we are told, is because a census was ordered and Jewish men were required to go with their families to return to the city of their origin. This is not the way a census was conducted in that time and place. The whole point was that Jesus had to be born in Bethlehem, so it could be argued that this wonky census was merely a literary device.

For more details, you may want to also ask /r/AcademicBiblical

The translation of the Greek word as "inn" is traditional, but there is a growing scholarly view that that is misleading if not simply incorrect; a better translation would be "guest room" in a private home. The entire picture of a public inn, like a first century Holiday Inn, probably puts us on the wrong foot to start with.

Here is fairly detailed and well-footnoted page that gives more details.