Boy, do I feel silly.
When I wrote yesterday, wondering about the meaning of "the sons of God" marrying the "daughters of men", I had failed to even consider today's verses!
Today's verse concerning the Nephilim tells us a number of things about these "sons of God" and "daughters of men".
First, their children were not ordinary human beings, so we can conclude that "sons of God" is not just an archaic way of referring to men. After all, if ordinary men marry ordinary women, their children are ordinary children, not superhumans. In fact, I think it is safe to say that the term "sons of God" does not actually refer to humans at all.
Second, the Nephilim were "men of renown" and "heroes of old". Based on these observations, I think it safe to say that their non-human fathers were superior to humans in at least some respects. This would explain why they could have children by ordinary human women and, through them, have such extraordinary offspring.
Third, despite the superior qualities of the Nephilim, the Bible seems to ndicate that their prescence on the earth was a bad thing, as was the union of their parents. We can deduce this by observing that the Bible has one verse concerning the Nephilim, and it is located between a divine proclamation that limits mankind's lifespan, on the one hand, and a description of the totally corrupt condition of mankind, on the other. We may be assured that, whoever the "sons of God" were, they shouldn't have been having children with the "daughters of men", and their union, according to Genesis, was just another symptom of the evil times that prevailed back then.
So these "sons of God" were (1) not human, (2) superior to humans in at least some respects, and (3) wicked. I only know of one type of being that fits all three of these descriptions, and that is a demon. So, if my assumptions are correct, demons were having children by human women during this time! This certainly underscores what a wicked place the earth was becoming!
Another point. The children produced in these unions were "heroes of old, men of renown". Does this sound to anyone like the Grecian myth about Hercules? Or Perseus? Or Orpheus? All of these characters were supposedly sons of one "god" or another. Is it possible that these characters from mythology really existed, and that demonic powers have encouraged various cultures to worship them as gods, while simultaneously having children by women from those cultures?
I don't really know the answer to any of this. But ultimately, I must keep in mind that the topic I have addressed yesterday and today is only a footnote to the main point of the passage: MANKIND HAD BECOME UNBEARABLY WICKED-SO MUCH SO THAT GOD FELT IT BEST TO DESTROY THE HUMAN RACE AND START OVER. And that situation was the inevitable consequence of one "tiny" sin in the garden of Eden. Sin destroys everything.
Then the Lord said, "My spirit will not contend with man forever, for he is mortal; his days will be a hundred and twenty years".
The Nephilim were on the earth in those days, and also afterward, when the sons of God went to the daughters of men and had children by them. They were the heroes of old, men of renown.
The Lord saw how great man's wickedness on the earth had become, and that every inclination of his heart was only evil all the time. The Lord was grieved that He had made man on the earth, and his heart was filled with pain. So the Lord said, "I will wipe mankind, whom I have created, from the face of he earth-men and animals, and creatures that move along the ground, and birds of the air-for I am grieved that I have made them."