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The priestly ministry was woven into the fabric of Jewish culture, as it was the God-ordained means under the old covenant for obtaining forgiveness of sins through animal sacrifice. Jewish believers who stopped participating in priestly rituals naturally came under fire from practicing Jews. How could they abandon the means God had given Israel to find forgiveness?
The answer, of course, is that God had appointed a superior and perpetual high priest of the new covenant, His very own Son, of whom all the previous priests only served to prefigure. The author of Hebrews points out why Jesus is fully qualified to serve as a high priest and why Jesus is superior to any before Him.
Every previous high priest was "beset with weakness," and each was "obligated to offer sacrifices for sins, as for the people, so also for himself" (5:3). Jesus, of course, had no such need to offer any sacrifice for Himself, as He was sinless. He was not only the perfect and superior high priest, but He was also the perfect and superior sacrifice. He offered up, not an animal, but Himself for our sins.
The author makes reference to verses in Psalms 2 and 110, both universally recognized by all Jews as messianic psalms that make reference to the Lord's future reign over the entire world. Both contain quotations of God speaking to God. In Psalm 2, the Father speaks to the Son---during the time when the Father will have installed His Son to rule the earth from Mt. Zion---saying, "Thou are My Son, today I have begotten Thee." In Psalm 110, the Father is again speaking to the One whom He has installed on Mt. Zion to rule the world. It begins with David prophetically saying, "The Lord says to my Lord" (110:1), and goes on to quote what God says to God, part of which is, "The Lord has sworn and will not change His mind, 'You are a priest forever according to the order of Melchizedek'" (110:4).
So it is indisputable that God revealed in the Old Testament that He would one day appoint the Messiah---the One who would rule over the world---to be a priest forever. Moreover, he would not be a priest after the order of Aaron, as were all previous priests, but after the order of a mysterious Old Testament man named Melchizedek, of whom we will read more about in chapter 7.
That Messiah and High Priest of whom Scripture foretold had been revealed, and so everyone in relationship with Him had obviously not lost anything relative to the benefits of a priesthood. And they would be foolish to go back to an inferior priestly system, one that was actually designed to point them to Christ.
When did Jesus offer up "both prayers and supplications with loud crying and tears to the One able to save Him from death" (5:7)? It must have either been when He prayed in the garden or from the cross, where we know that our great High Priest and holy sacrifice cried out with a loud voice, "My God, My God, why have You forsaken Me?" (Matt. 27:46). The author writes that Jesus "was heard because of His piety" (5:7). Perhaps this is a reference to Him being "saved from death" (5:7) by means of His resurrection. Our High Priest receives what He prays for, which should fill us with confidence.
If Jesus was sinless, why did the author write that Jesus "learned obedience from the things which He suffered" (5:8)? The author could not have meant that Jesus learned to become obedient by suffering the consequences of disobedience, but rather that He learned from experience the cost that is paid by those who are obedient to God. It cost Jesus His life. But His sufferings resulted in His complete perfection, not making Him morally perfect (since He already was), but making Him the perfect Savior and High Priest.
Finally, notice that Jesus "became to all those who obey Him the source of eternal salvation" (5:9). This is just one more indication that there is a correlation between holiness and heaven because there is a correlation between belief and behavior.