My obsession with eunuchs continues. I ask about this tablet (here's the British Museum's entry, and here's the Wikipedia article) because the translations I've seen describe Nabu-sharrussu-ukin as Nebuchadnezzar II's chief eunuch and mention another eunuch named Arad-Banitu. But I've also heard that the Akkadian and Hebrew words for "eunuch" (respectively "sa resi" and "saris") were also often used to describe government officials in general. Have people confirmed that this guy actually underwent the operation; if so, how did they determine it? I find Nabu-sharrussu-ukin particularly interesting because, in contrast with Assyria and Persia, eunuchs don't seem very well attested in Babylon (though the Talmud apparently describes the biblical Daniel and his companions as eunuchs).
For the Assyrian/Babylonian term:
It is possible that the term means different things in different contexts. So the term might mean 'eunuch' or 'high official' or 'high official who is a eunuch'.
At least for the Neo-Assyrian period, we have evidence that the sha reshi officials did not marry and did not have children (there are texts dealing with their inheritance and others discussing who is responsible to provide their funerary offerings as they don't have children). They also seem to have been depicted beardless in Neo-Assyrian art.
Furthermore, there is an omen that says 'May your semen dry up like that of a sha reshi who cannot beget' and, in the Laws, being turned into a sha reshi is the punishment for adultery.
So I think we can safely say that at least some sha reshi are eunuchs, at least in Assyria.
In the Babylonian material, on the other hand, there is no clear evidence to prove or disprove that they were eunuchs.
As far as I am aware (I did check in the Babylonian Prosopography database), the tablet in the British Museum that you mention is the only Mesopotamian text mentioning 'your' Nabû-šarrūssu-ukīn, so there is no other evidence regarding whether this specific individual was a eunuch, other than his title mentioned in this one tablet.
I am by no means an expert on the Biblical evidence, but it seems more accepted that the Hebrew term means 'eunuch', and in the Bible, the saris are also mentioned, for example, guarding the women's quarters.
Here is some literature that might help you further (sorry, everything is in really obscure publications):
A. K. Grayson (1995): 'Eunuchs in Power. Their Role in the Assyrian Bureaucracy.' In: Dietrich & Loretz (eds.): Vom Alten Orient zum Alten Testamen. Festschrift für Wolfram, Freiherr von Soden. (=AOAT 240). Kevelaer: Butzon & Becker. Pp. 85-98.
K. Deller (1999): 'The Assyrian Eunuchs and their Predecessors.' In Watanabe (ed.): Priests and Officials in the Ancient Near East. Heidelberg: Winter. Pp. 303-311.
H Tadmor (1995): 'Was the Biblical saris a Eunuch?' In: Zevit, Gitin & Sokoloff (eds.): Solving Riddles and Untying Knots. Biblical, Epigraphic, and Semitic Studies in Honor of Jonas C. Greenfield. Winona Lake: Eisenbrauns. Pp. 317-325.