The “inn” where there was no room was likely actually a family member’s home. Luke uses the word “inn” to describe where Mary and Joseph were turned away, but he also uses the word “inn” to describe where the good Samaritan took the Israelite found on the road. In the case of Mary the word Luke used was κατάλυμα (katalyma) which can be interpreted either as an inn or as a “guestchamber.” The word for “inn” Luke used in the story of the good Samaritan was a very different one. It was πανδοχεῖον (pandocheion) which is translated as a public house for strangers. Obviously if Luke had wanted to let us know that Mary and Joseph were turned away from a public inn he had the language with which to do it. (Source)