I believe Paul is teaching that in this new covenant all matters are fulfilled (completed not abolished) spiritually. There is therefore no physical temple & no point to Christ offering physical sacrifices as the sacrifices in this new covenant are now spiritual.
The support for a spiritual vs physical temple is found in:
Revelation 21:22 I did not see a temple in the city (no physical temple), because the Lord God Almighty and the Lamb are its (spiritual) temple.
Ephesians 2:20-22 Built on the foundation of the apostles and prophets, with Christ Jesus himself as the chief cornerstone (of this spiritual temple). 21 In him the whole building (spiritual temple) is joined together and rises to become a holy temple in the Lord.
John 2:19-21 Jesus answered them, "Destroy this temple, and I will raise it again in three days (as chief cornerstone of the spiritual temple)." 20 They replied, "It has taken forty-six years to build this temple, and you are going to raise it in three days?" 21 But the temple he had spoken of was his body.
1Corinthians 6:19 Do you not know that your bodies are temples of the holy spirit who is in you, whom you have received from God? You are not your own;
We are joined together as a spiritual temple in the Kingdom of Heaven.
In Hebrews 8:4 Paul is making the point that in this new covenant the physical sacrifices are replaced by spiritual sacrifices. In the same chapter Paul (Hebrews 8:6-13) states we now have a new and superior covenant and high priest.
Physical sacrifices were never the point. It was our obedience to His Holy Laws & Commandments which is considered our spiritual sacrifice.
Micah 6:6-8 With what shall I come before the lord and bow down before the exalted God? Shall I come before him with burnt offerings, with calves a year old? 7 Will the lord be pleased with thousands of rams, with ten thousand rivers of olive oil? Shall I offer my firstborn for my transgression, the fruit of my body for the sin of my soul? 8 He has shown you, O mortal, what is good. And what does the lord require of you? To act justly and to love mercy and to walk humbly with your God.
1Samuel 15:22 But Samuel replied: "Does the lord delight in burnt offerings and sacrifices as much as in obeying the lord? To obey (our spiritual sacrifice) is better than (physical) sacrifice, and to heed is better than the fat of rams.
Lets conclude with a verse just 6 verses after the verse you cited (Hebrews 8:4).
Hebrews 8:10 This is the covenant I will establish with the people of Israel after that time, declares the Lord. I will put my laws in their minds and write them on their hearts. I will be their God, and they will be my people.
In Hebrews 8 Paul teaching this new covenant with greater promises & a superior High Priest & our spiritual sacrifice governed by the gift of a loving heart to keep His Laws.