Q. How can the book of Hebrews say that Melchizedek, the priest who blessed Abraham, was “without father or mother” and “without beginning of days or end of life”? Wasn’t he human?

Here’s what I say about this in my study guide to Deuteronomy and Hebrews, where I note that the author of Hebrews talks about Melchizedek in the third of the four messages or sermons that make up the book:
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This message is based primarily on Psalm 110, but in it the author characteristically draws on other Scriptures for support, in this case the story in Genesis that describes who Melchizedek was.
The author first translates the word Melchizedek, explaining that it means “King of Righteousness.” Melchizedek was most likely not a given name, but an honorary title of the Jebusite kings who formerly ruled in Jerusalem, including the one in the Genesis story who greeted Abraham. (A similar example of an honorary title is the name Pharaoh that was given to all the rulers of Egypt.)
After the Israelites conquered Jerusalem, their own kings took over the title Melchizedek. Since the Jebusite kings had been priests, the Israelite kings also assumed an honorary role as priests and interceded for the nation in prayer. But they were not allowed to offer sacrifices; this was reserved for the descendants of Aaron under the law of Moses.
The author next explains that King of Salem (that is, of Jerusalem) means “King of Peace.” By translating these two terms, the author identifies Jesus, who is a priest in the order of Melchizedek by virtue of being the Messianic king of Jerusalem, as someone who helps people become righteous before God and so find peace with God.
Now come some more significant details—or rather, a significant lack of them. The Hebrew Scriptures usually introduce a new figure into their narratives by describing the person’s parentage and ancestry. They usually also report when a figure dies. But the book of Genesis doesn’t do either of these things in the case of Melchizedek.
This allows the author of Hebrews to observe that, when considered only in light of what the Scriptures say about him, Melchizedek seems to have no origin or ending. He appears to “remain a priest forever.” In this way he “resembles the Son of God,” and this allows him to serve as an earthly representation of the Messiah. This is why the Lord chose to name him as the head of the order of priests to which the Messiah (represented in Psalm 110 by the Davidic king) would belong.
This is a classic example of the author’s typological method, which is based on the understanding that transcendent spiritual realities are reflected in earthly replicas. A little later in this message the author makes the basis of this method explicit, noting how the earthly tabernacle had to be modeled after the heavenly pattern Moses was shown. The Greek word is typos, the source of the English word type, and so this interpretive method is known as typology.
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To summarize what I say in the guide, the author of Hebrews is able to establish a connection between Melchizedek and Jesus by considering Melchizedek in light of what the Scriptures say about him (that his title means “king of righteousness” and that he was king of Salem = “peace“), but only in light of what the Scriptures say about him, not what they don’t say. Since the details of his parentage, birth, and death aren’t reported, this allows an even stronger typological connection to Jesus, who has a permanent priesthood “on the basis of an indestructible life.”
In other words, the key to understanding how the Bible could say that Melchizedek was “without father and mother” and “without beginning of days or end of life” lies in appreciating the distinctive typological method of the book of Hebrews.