But I've recently been reminded that the Bible also refers to nephilim as "giants", and that the original Hebrew word, nefilim, apparently means "the fallen ones".
I believe you are referring to yet another, enigmatic, anomalous and ambiguous half story that is left in mid-air.
There were giants in the earth in those days; and also after that, when the sons of God came in unto the daughters of men, and they bare children to them, the same became mighty men which were of old, men of renown.Genesis 6:4 KJV
Other bibles use the word "nephilim" such as:
The Nephilim were on the earth in those days, and also afterward, when the sons of God came in to the daughters of man and they bore children to them. These were the mighty men who were of old, the men of renown.Genesis 6:4 ESV
Are the nephilim "the sons of god" or are they "offspring" of the sons of god who are the "mighty men of old, men of renown".
"Giant" and "men of renown" could simply mean status and not stature? It could mean that they were highly intelligent beings?
This is a bit more to the story:
When people began to multiply on the face of the ground, and daughters were born to them, the sons of God saw that they were fair; and they took wives for themselves of all that they chose. Then the Lord said, "My spirit shall not abide in mortals forever, for they are flesh; their days shall be one hundred twenty years." The Nephilim were on the earth in those days—and also afterward—when the sons of God went in to the daughters of humans, who bore children to them. These were the heroes that were of old, warriors of renown.
This is interesting as it mentions the "nephilim" as the descendants of Mesopotamian lord/ gods known as the Anunnaki. Are these the "men of old" and the "men of renown" that they are speaking of?
The Lord said to Moses, "Send men to spy out the land of Canaan, which I am giving to the Israelites" ... So they went up and spied out the land ... And they told him: "... Yet the people who live in the land are strong, and the towns are fortified and very large; and besides, we saw the descendants of Anak there." ... So they brought to the Israelites an unfavorable report of the land that they had spied out, saying, "The land that we have gone through as spies is a land that devours its inhabitants; and all the people that we saw in it are of great size. There we saw the Nephilim (the Anakites come from the Nephilim); and to ourselves we seemed like grasshoppers, and so we seemed to them."
— Numbers 13:1–2; 21; 27–28; 32–33. N R S V.
As can bee seen , here they make it clear that these being are of some size.
I mentioned on another thread that these Mesopotamian Lords didn't seem particular who they mated with. And maybe this is why they were known as "fallen".
And the book of Enoch goes into more detail telling us that some of these "fallen" were imprisoned for what was considered taboo.