This verse, and the verses surrounding it, is speaking of who is to guard the Ark of the Covenant, not a general qualification for priesthood.
kbertsche writes:
According to the Old Testament, priests of Israel were exclusively from the tribe of Levi (who was one of the sons of Jacob):
Num 3:6-10
Bring the tribe of Levi near, and present them before Aaron the priest, that they may serve him.
... You are to assign the Levites to Aaron and his sons; they will be assigned exclusively to him out of all the Israelites.
So you are to appoint Aaron and his sons, and they will be responsible for their priesthood; but the unauthorized person who comes near must be put to death.