The eunuch was an individual who was NOT heterosexual. That is, the eunuch did not engage in sexual intercourse with members of the opposite sex.
In this regard, eunuchs therefore were the custodians of the consorts of the king, because they (the eunuchs) were not a threat. For example, in the Book of Esther we find Hegai, one eunuch of King Ahasuerus, who was in charge of "the women" of the court (Esther 2:3), and then there was Shaashgaz, who was the eunuch in charge of "the concubines" of the king (Esther 2:14). There was yet another eunuch assigned to attend and serve Queen Esther, and his name was Hathach (Esther 4:5). In other words, these eunuchs were trusted agents of the king who were not to have any sexual relations with the king's female consorts.
The expectation of sexual purity here however was not based on an honor system of self-restraint, but on sexual castration: thus these eunuchs were not physically able to have sexual intercourse with the female consorts of the king. As gilded males, they were therefore "dried trees" that could produce no fruit or offspring (Isaiah 56:3 compared to Isaiah 56:5). Therefore they were not a threat to the king's harem because they were castrated males. So these are the eunuchs to whom Jesus was referring "who were made eunuchs by men," since their male genitalia are removed: thus their sexual identity is altered surgically.
In other words, the idea here is that sexual orientation is changed by the surgical removal of the genitalia. This removal of the genitalia does not make the eunuch homosexual (although there is ample literature that eunuchs were known as homosexuals in the ancient world -- for example, please click here), but the castration as a minimum erased any heterosexual identity of the male concerned.
Now there are others who are born eunuchs. That is, there are some people who are born with disfigured parts of their bodies (cf. 1 Chronicles 20:6), and those with disfigured genitalia are hermaphrodites, for whom their sexual identity is not heterosexual by definition, since their sexuality is medically termed intersexual (that is, their sexuality is literally ambiguous). These people are therefore born transsexual.
Finally, there are those who have chosen to dedicate their sexuality to God (irrespective of sexual orientation), and therefore have chosen to abstain from any sexual behavior, and these are eunuchs for the Kingdom of God. The Apostle Paul advocated for such dedication in 1 Cor 7:7-9.
Before concluding, the mention of homosexuality is relevant, since eunuchs and homosexuality were not unrelated topics in the ancient world as noted in the previous paragraphs. That is, the idea here is that one can become a homosexual because of the acts of other people -- for example, the heterosexual who is castrated in order to be a eunuch may in fact later become a homosexual. That is, castration physically alters sexual identity. However, the idea of "castration" can also include the alteration of sexual identity through exposure to homosexual behavior (even though the male is not castrated physically). That is, a male who is born heterosexual can become homosexual because of the acts of other people (to include sexual abuse). Thus this male is made a "eunuch" by others (even through he is not physically castrated).
Conversely, one can be born with ambiguous genitalia and therefore will not be heterosexual, but "intersexual" (transgender). These people are born eunuchs, since any identity of heterosexuality is "castrated" by the physical deformity of their genitalia. These people are therefore born as eunuchs, because they are not heterosexuals in the medical physiological sense. The suggestion now is whether someone can be born homosexual, since sexual identity (as noted in the previous paragraph) does not necessarily depend on the mutilation of genitals. The following graph best depicts the parallels of these concepts.
In summary, there are the eunuchs made by men, who are either castrated males (that are not heterosexual) or homosexuals, who are not castrated -- in either case, their sexual identity as heterosexuals is "cut away." Secondly there are also those who are born eunuchs, whose sexual orientation is ambiguous by birth -- that is, their sexual ambiguity may or may not depend on deformed genitalia. And finally, there are those who commit themselves to be eunuchs for the Kingdom of God, and who practice complete sexual abstinence, and who, like the apostle Paul (or even the castrated eunuchs in Isaiah 56:3-5), dedicate themselves to the service of God's glory.
In conclusion, Jesus calls us to recognize that there are eunuchs for the Kingdom of God whose sexual identity (whether heterosexual or otherwise) is consecrated to the complete glory of God. These eunuchs would also include by invitation those who were made eunuchs by the agency of man and those who were made eunuchs by the agency of birth.