While only those of the Levite clan could serve as priests in the Old Testament, things change in the New Testament when God takes the non-levitical Jesus and makes Him a king/priest forever. As Hebrews says, Jesus “has become a priest, not on the basis of a legal requirement concerning bodily descent, but by the power of an indestructible life” (He 7:16).
And while we are not Jesus the forever king/priest, Peter still calls us, “a chosen race, a royal priesthood, a holy nation, a people for his own possession, that you may proclaim the excellencies of him who called you out of darkness into his marvelous light” (1 Pe 2:9). The book of Revelation agrees, saying that Jesus has “made us a kingdom, priests to his God and Father” (Rev 1:6). Therefore, nothing and no one can stand in the way of us accessing the presence of God. As one scholar notes,
we occupy so high a position that no man can be higher in this life: as a “priesthood,” a body that is made up entirely of priests, no man stands between us and God, and as a body of “royal” priests no man stands over us in our relation to God. The adjective as well as the noun reveal in a double way the exaltation of our position and our function, the constant direct, immediate contact with God. (Lenski, R. C. H. The Interpretation of the Epistles of St. Peter, St. John and St. Jude. Minneapolis, Augsburg Publishing House, 1966, p. 100.)
Our genetics no longer determine our ability to approach the holy of Holies behind the curtain—we do not need to be Levites to enter the sacred space and the author of Hebrews invites us to do so freely:
Therefore, brothers, since we have confidence to enter the holy places by the blood of Jesus, by the new and living way that he opened for us through the curtain, that is, through his flesh, and since we have a great priest over the house of God, let us draw near with a true heart in full assurance of faith, with our hearts sprinkled clean from an evil conscience and our bodies washed with pure water. (He 10:19-22)
Jesus has made a way for us. He is both the king and the high priest for all eternity and He has invited us to join His kingdom by operating as His royal priesthood. And so when we accept salvation, we become more than we’ve ever been before. We are a royal priesthood that is able to access the intimate presence of God because of the king/priest Jesus—the new Melchizedek. And as Jesus welcomes us into the priesthood, we are given the ability to do exactly what priesthoods do and perform in the “office for mediation between the divine and human.” (Carver, Daniel E. “Priesthood.” Lexham Theological Wordbook. Lexham Bible Reference Series. Editor Douglas Mangum et al. Bellingham, Lexham Press, 2014.)