Actually, there are two Greek words typically translated into English as "inn." One (a recent creation) means close to what we mean by "hotel" - a place and business where travelers can spend the night. The ancient Greeks were probably the first to invent these, and it greatly expanded during Roman times. They were found in all major cities and at times at junctions of major Roman highways. They would have an innkeeper, a registration desk, etc. There would be no such inn in the tiny, out of the way village of Bethlehem. This word appears only once in the Bible, in the parable of the Good Samaritan, where the wounded traveler is taken to an inn to recover.
But that's not the word found in the Christmas narrative. That word simply means a place to sleep. That's all. It could equally be translated "bedroom," although the connotation is a place where guests could sleep. There would be no innkeeper, no registration desk, no vacancy / no vacancy sign outside - just SOME PLACE where a guest could lay down a bed roll. Important in a day when hotels were rare (and expensive). Houses of common people back then generally consisted of just one room, with no fireplace.... and no kitchen or bath (those where outside).... and priority was given not to the family but to the animals! They needed to be kept safe and contained at night. In the winter, the family slept with them because they generated much appreciated heat. Yes, the room was kept very clean, it was constantly cleaned, since this was space shared. When the weather was warmer (and the animals, hum, perhaps not smelling that great), the family simply slept on the roof (cooler, better air).
We have no clue when Jesus was born... but it seemed the house they asked to stay at at that time (probably that of a relative) didn't have any space in the usual sleeping space. That's all the text actually says. Was this during a warmer season and the roof was crowded? Were they instead sleeping downstairs, with the animals? That "fits" better with our image. Or was it cooler and the Holy Family was sent up to the roof, requiring blankets but still hospitable (and private)?
What about that manger? It was, after all, a feeding trough typically made of stone (but sometimes wood)? Doesn't that prove the weather is nice and the Holy Family is down there with the animals? Maybe.... but it is possible the manger was simply containing hey needed for those bedrolls and not for the animals at all. All hey containers were called "mangers" - whether animals ate from them or not. And it's entirely possible the typical crib was a manger... one supplied as the nursery.
It's unlikely the host family were the meanies we've made them out to be ... although they likely didn't give up THEIR space (and should have), they lacked the hospitality Jews were known for, but they did gave the family a place to stay... and some privacy.
What about the earliest tradition, that Jesus was born in a cave? Well, caves were often used as barns but seldom as homes. It COULD be (especially in Winter) that Mary and Joseph were given that space in which to live while in
Bethlehem... in winter, it was common to live with the animals. This would diminish the "no room in the inn" comment, however... but perhaps that ONLY means that they were not living in (or on) a house (the usual places to sleep).
By the way, there's nothing in the text (or early tradition) that suggests Mary and Joseph had just arrived in Bethlehem. Most scholars think they would have been there for several weeks before Jesus was born. It's thus possible this sleeping arrangement was made long before it was Mary's time to deliver. The Bible says "WHILE they were THERE...." nothing implying Jesus was born right away.
