The Nazirite vow was a means of extending the guarding role of the priesthood to an Israelite - either male or female - for the purpose of holy war. It was a sort of "priestly knighthood."
The vow is a miniature of Israel's sojourn in the wilderness - an emptying and a humbling followed by a filling and a glorification. The "Covenant head" is empty and comes back with "bridal hair." The grapes of Canaan are refused until the vow is complete and the Land is taken.* The idea goes back to the two trees in the Garden: the second tree temporarily forbidden for the sake of the humbling of Adam and the glorification of the Bride. Phil 2:5-11 also follows this pattern of emptying and filling.
The closest thing under the New Covenant is believer's baptism (which is also for both men and women). Believers abstain from "kingdom privileges" (food, alcohol, sex) temporarily for the sake of priestly war (1 Corinthians 7:5).
*Also, notice that no wine is drunk before God between Melchizedek's blessing of Abraham and the Last Supper (the Greater Melchizedek). The entire period of the Abrahamic Covenant was a priestly humbling, a temporary abstinence from kingly food for the maturation and qualification of humanity for Adamic rule.