Making themselves eunuchs should be interpreted symbolically or functionally as wounding yourself in the stones was a sin in Judaism (Deuteronomy 23:1). It means taking an ascetic/monastic life.
23:1 A man with crushed or severed genitals may not enter the assembly of the Lord.
While your question title refers to Christian groups, you mention how Jesus speaks of the practice as if it were somewhat common. That would mean exploring Jewish groups at the time.
As to who at the time of Jesus He might be referring to, the Essenes are the most likely candidate. Many of the Essenic communities practiced celibacy though this was not required by the Teacher of Righteousness when he founded the community and their practices approximately 200 years before Jesus. Although he did require them to lead pure lives, in the Damascus Covenant, he states:
if members of the community happen to be living in encampments, in accordance with a usage which obtains in this country, and if they marry and beget children, they are [in such matters] to follow the precepts of the Law [Torah] and the disciplinary regulations therein prescribed for the relationship of husband to wife and of father to child. (vii 6a-9)
It later became common for Esseneic camps to practice celibacy. Pliny, Josephus, and Philo all mention it.
By the western shores [of the Dead Sea], but away from their harmful effects, live a solitary people, the Essenes, wonderful besides all others in the world, being without any women and renouncing all sexual desire, having no money, and with only palm trees as companions. (Pliny, Natural History 5:18:73)
Furthermore they eschew marriage because they clearly discern it to be the sole or the principal danger to the maintenance of the communal life, as well as because they particularly practice continence. For no Essene takes a wife, because a wife is a selfish creature, excessively jealous and an adept at beguiling the morals of her husband and seducing him by her continued impostures. (Philo, Hypothetica 11:14)
For there are three philosophical sects among the Jews. The followers of the first of which are the Pharisees, of the second, the Sadducees, and the third sect, which pretends to a severer discipline, are called Essens. These last are Jews by birth, and seem to have a greater affection for one another than the other sects have. These Essens reject pleasures as an evil, but esteem continence, and the conquest over our passions, to be virtue. They neglect wedlock, but choose out other persons children, while they are pliable, and fit for learning, and esteem them to be of their kindred, and form them according to their own manners. They do not absolutely deny the fitness of marriage, and the succession of mankind thereby continued, but they guard against the lascivious behavior of women, and are persuaded that none of them preserve their fidelity to one man. (Josephus, The Wars of the Jews II 8.2)
However, Josephus also notes that some groups of Essenes married:
Moreover, there is another order of Essens who agree with the rest as to their way of living, and customs, and laws, but differ from them in the point of marriage, as thinking that by not marrying they cut off the principal part of human life, which is the prospect of succession. Nay, rather, that if all men should be of the same opinion, the whole race of mankind would fail. However, they try their spouses for three years, and if they find that they have their natural purgations thrice, as trials that they are likely to be fruitful, they then actually marry them. But they do not use to accompany with their wives when they are with child, as a demonstration that they do not many out of regard to pleasure, but for the sake of posterity. Now the women go into the baths with some of their garments on, as the men do with somewhat girded about them. And these are the customs of this order of Essens. (Josephus, The Wars of the Jews II 8.13).