The author of Hebrews seems to think Melchizedek has no beginning or end. This is an important part of the argument, since the author is arguing Jesus is part of a higher order of priesthood that does not depend on the Mosaic law and is not subject to human mortality, and that Melchizedek is an example of this higher order.
The author seems to think Melchizedek is literally immortal.
Without father or mother or genealogy, without beginning of days or end of life, like the Son of God, he remains a priest for all time.
- Hebrews 7:3
In the case of the Levites, mortal men collect the tenth; but in the case of Melchizedek, it is affirmed that he lives on.
- Hebrews 7:8
And this point is even more clear if another priest like Melchizedek appears, one who has become a priest not by a law of succession, but by the power of an indestructible life. For it is testified:
“You are a priest forever in the order of Melchizedek.”
- Hebrews 7:15-16
The most straightforward reading of the above verses is that the author thinks Melchizedek is immortal, and that is what makes his priesthood superior to the Levitical priesthood. Furthermore, Jesus is like Melchizedek precisely because he has an indestructible life, which the author backs up with an old testament quote to the same effect.
Finally, Jesus' salvific role is tied specifically to being immortal, i.e. like Melchizedek.
But because Jesus lives forever, He has a permanent priesthood. Therefore He is able to save completely those who draw near to God through Him, since He always lives to intercede for them.
- Hebrews 7:24-25
In which case, is Melchizedek some kind of immortal human? Is he still alive today? Was he around when Adam and Eve were created?