The Archbishop of Canterbury said, recently, "I should think so," regarding the question, "...historically and factually true?" Let's take a more detailed look.
There may be a common source for the birth narratives but the two Gospels do not agree. Did his mother, who survived him, give any account of the birth to the curious? Who knows, but from a scriptural point of view they were constructed in various places to connect to the prophecies given in the Hebrew Bible. Plus, the further away the Christians get the more the believers are likely to be curious about his origins.
Micah 5:2 says the Messiah would be born in Bethlehem.
If Jesus thought he came from Bethlehem, would not more have been made of this (given its central prophetic role?). Everyone took it that he came from Galilee. In John 1:45-46 Peter of Nathaniel asks if any good can come out of Nazareth. Nicodemus was told to search the scriptures and see that no prophet comes out of Galilee (John 7:43), on account of Jesus coming from there.
Luke says that Joseph took his pregnant wife 70 miles to Bethlehem for the Roman enrolment for taxation, Joseph being a descendent of King David, Bethlehem being the city of David. Acts 5:37 claims an enrolment in Judaea, but (if so) this required only men to go to their own city for enrolment. Luke says it applied to the whole empire, however. The mind boggles at that.
The trek to Bethlehem does not have credibility. The Romans kept extensive records, but there is no evidence of such a tax based census, and the one that might have been it has a date at 6 CE, some 6 to 10 years after Jesus was born. Nor would the Romans have organised such movements of masses of people, in Palestine never mind the whole empire, given their concern about security. Nor did they care about Jewish origins - they only wanted to know about what the Jews owned.
Matthew puts the trip the other way around. The three of them start in Bethlehem and they end up in Nazareth. Now why did they want to do that? This time, though, they moved via Egypt, in order to fulfil another prophecy (Hosea 11:1). So that is a 450 miles detour to Egypt and back, which could not have been much fun for a newly born baby named Yehoshua (Yahweh saves) and mother. Perhaps Matthew lived in Alexandria and gave the birth account some local colour.
Factual and historical Bethlehem is not. Nor is the massacre of the innocents (no mention of one anywhere by Josephus), the star, nor the virgin birth.
In those days stars were considered to be on the roof of the world, like lights. We have different evidence. Halley's Comet appeared 12 CE - remember that Comets do not apear to move. As regards the Magi, or astrologers, well it might have been Isaiah 60:3 but the Romans had thre Magi accompanying Tiridates of Parthia to go and worship the baby Nero as Mithras, and so to this Messiah may have had the same recognition as a demonstration for such Gentiles. They must have been a bit thick, these Jesus following Magi, losing sight of such a star and then making the effort to ask Herod, of all people. It's a multi-sourced fable. The notion of "His star" is a connection with Levi 19:3, in The Testament of the Twelve Patriarchs. Numbers 24:17 in the Septuagint also uses the star myth, and the Talmud has a star rising in the east and seven others fighting. When Mithras was born as star fell out of the sky and Magi followed that. It is simply daft talking about "scientific theories" about a star, as the Archbishop of Canterbury did when interviewed by Simon Mayo. Magi are not "professionals" looking for signs, and certainly would find nothing in stars. It is funny that astrology can be dismissed as bunk, when astrology as explanation and comfort falls into the same category as much other religion.
The Archbishop finds the Virgin Birth increasingly meaningful. It is, directly, a mistranslation of Isaiah 7:14, that a damsel would conceive and bear a son. Being fatherless derives from the mistranslation, not precedes it - unless of course there was a rumour about illegitimacy (but why would there have been?). Myths abound about Gods not having human fathers, of course. Plato, Pythagoras, Augustus and Alexander the Great all have virgin birth stories attached to them, and for Buddha in Buddhism too. Yet, in some contrast and contradiction to this Virgin Birth doctrine, Joseph is given as a descendent of David (which is a nonsense on its own terms, and given even our difficulties with genealogy is hardly likely to be evidenced) - the ancestor more for the Jews perhaps, and virginity more for Gentiles.
The gospels are not principally histories. Just because we deal with them as if biography and "history-like" does not make them biographies or histories. We cannot jump into an imaginary world and declare fact and history: that really does take postmodernism to the extreme. They are theologies of the early Churches written in biographical historical form, and do not even work from primary sources. There is history to be found in them, by sifting, with great difficulty, but the least historical and factual parts of two are the birth narratives.
None of this is news to the Archbishop. Factual and historical? I should think not. Or, in simpler language: no.
So what of an actual and unknown to the world birth (history does not go backwards). Galilee is one thing, but Nazareth is another. Neither the Talmud nor Josephus mentions Nazareth as even existing, but archaeological work suggests it did and was a very small village. So was it the place the evangelists described? It would not have had a synagogue - but there was one in Capernaum (Mark 1:21). It is usually taken that Nazarene means someone from Nazareth, but why would such a person not be a Nazarethene? It looks like Nazarene may have another meaning entirely, which could be the group name, and one that was less significant over time and even to be concealed. Matthew (2:23) adds a prophecy that never existed, that the Messiah would be called a Nazarene.
If Jesus was not born in Nazareth (which is still possible), then there is another contender, and that is Caparnaum, known as his own city (Mark 9:1). Jesus was arguably based there, on the north west shore of Lake Tiberius (the Sea of Galilee) and where the fishermen were active. There was a centurion there (Matthew 8:5). Jesus performed miracles in Capernaum (Mark 11:23).
There he grew up, speaking Aramaic and Koine (probably - a rough Greek) and caught the last days fever of his time, with the religious prophets of his day, healing so many as they believed that the unwell had demons, and the poor had more than most. The demons had to go to be ready for the Kingdom, and those people had to sin no more while they waited. Those demons wore you down and were responsible for your death too, death being caused by sin. Know where this is heading?
Meanwhile, how was Lord Krishna supposed to be born? His parents were journeying to pay taxes, and he was born in a manger. There were shepherds visiting the birth. The new born baby escaped infanticide and they went into exile.
Campbell, Steuart (1996), The Rise and Fall of Jesus, Edinburgh: Explicit Books.