Genesis 6:1--4 tells about the "sons of God" marrying whomsoever they pleased, and the Nephilim---"the heroes of old, men of renown"---were begotten then. How is this to be interpreted?
Dr John Millam, on the Reasons to Believe website, has this to say on the Nephilim (about 1/2 way through, the section entitled The Sons of God/Giants):
Historically, the most popular way of understanding this passage was that the "sons of God" refer to angels or to the descendents of the godly line of Seth, Adam's son. The translators of the Septuagint, however, biased interpretations toward one standpoint by substituting "angels" for "sons of God". (The sons-of-God-as-angels view was fashionable in Jewish circles at that time. It appears in the apocryphal Book of Enoch, so the translators of the Septuagint were simply reflecting popular thinking.) This helps explain why the angel interpretation held exclusive dominance in the church for the first two centuries. A second example can be found in this same passage where the Septuagint translated Nephilim (literally "fallen ones") as "giants" (Greek gigos). As a result, the view that the Nephilim were "giants" was rampant in the early church.
Most people I have read on the matter agree that it is highly uncertain who these "Nephilim" actually were, and as a result, speculation abounds. In my thinking it seems problematic to consider that angels, who are purely spirit beings could mate and produce offspring with humans who are a distinct creation of body into which God breathed spirit.
From the wikipedia article:
Some individuals and groups, including St. Augustine, John Chrysostom, and John Calvin, take the view of Genesis 6:2 that the "Angels" who fathered the Nephilim referred to certain human males from the lineage of Seth, who were called sons of God probably in reference to their being formerly in a covenantal relationship with Yahweh (cf. Deuteronomy 14:1; 32:5); according to these sources, these men had begun to pursue bodily interests, and so took wives of the daughters of men, e.g., those who were descended from Cain or from any people who did not worship God.
While the ca. 6th century The Conflict of Adam and Eve with Satan (a Christian pseudepigraphical work found in Ge'ez) says:
Certain wise men of old wrote concerning them, and say in their [sacred] books, that angels came down from heaven, and mingled with the daughters of Cain, who bare unto them these giants. But these [wise men] err in what they say. God forbid such a thing, that angels who are spirits, should be found committing sin with human beings. Never, that cannot be. And if such a thing were of the nature of angels, or Satans, that fell, they would not leave one woman on earth, undefiled... But many men say, that angels came down from heaven, and joined themselves to women, and had children by them. This cannot be true. But they were children of Seth, who were of the children of Adam, that dwelt on the mountain, high up, while they preserved their virginity, their innocence and their glory like angels; and were then called 'angels of God.' But when they transgressed and mingled with the children of Cain, and begat children, ill-informed men said, that angels had come down from heaven, and mingled with the daughters of men, who bare them giants.
Which seems to make sense to me. Both wikipedia articles contain numerous references which may be of interest for further research and reading.
A second reference to Nephilim occurs in Numbers 13:32-33, but since all of mankind was destroyed in the flood of Gen 9 ff, I don't see how they could be of the same bloodline. So that suggests that Nephilim might be more of a general than specific descriptive term. It may be significant that both groups of men are described as "giants" (note also that Goliath and his kin, "descendants of Rapha in Gath" were described as having six fingers and six toes, which indicates some sort of genetic deformity).
The New Testament teaches us that the Sons of God are the righteous.
For as many as are led by the Spirit of God, they are the sons of God.
But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name
1 John 3:1
Behold, what manner of love the Father hath bestowed upon us, that we should be called the sons of God: therefore the world knoweth us not, because it knew him not.
That last bit is instructive. "The world knoweth us not, because it knew him not." It hearkens back to the Savior's words:
John 15: 18-19
18 If the world hate you, ye know that it hated me before it hated you.
19 If ye were of the world, the world would love his own: but because ye are not of the world, but I have chosen you out of the world, therefore the world hateth you.
So we see that the Sons of God are chosen out of the world and are different from "the world." So if the Sons of God married daughters of men, (ordinary women "of the world" who were not living in righteousness,) the children born to them may have been great and powerful, but without a righteous mother to raise them in the truth they would have fallen away from God. This was always a big deal in Old Testament times. In the Law of Moses, marrying outside the covenant was forbidden, and Solomon choosing to ignore this prohibition is explicitly blamed for the decay and downfall of his kingdom.
Others have noted the two keys theories: (a) angels mated with human women, or (b) descendants of the Godly line of Seth intermarried with the sinful line of Cain.
The Bible uses the phrase "sons of God" to refer to angels in Job 1:6, 2:1, and 28:7. It uses the phrase to refer to Christians in Matthew 5:9 (sort of) and Galations 3:26. It appears to use both meanings in the same sentence in Luke 20:36. Note that the exact meaning of the phrase in all these cases is debatable. I suppose advocates of the angels theory have an advantage here in that the phrase in Genesis and the phrase in Job are the exact same Hebrew words ("bni Elohim"), while the statements in the New Testament are in Greek, not Hebrew, and thus different words. Thus the connection is a little more tenuous.
Personally, I find both theories problematic.
RE angels: Is it physically possible for angels to interbreed with humans? I suppose an advocate of this theory could reply, Who can say? How do we know whether it's physically possible or not? In many cases in the Bible angels appeared to people as human males. Often people didn't realize they were not ordinary human males until the angel told them he was not or revealed himself in some way. If angels can appear as human males with apparently-functioning arms and legs, mouths that can speak, etc, perhaps they also have or can have other parts that function like a human male.
Yes, Jesus said that angels do not marry. But does this mean that they don't have sex, or that in Heaven there is some other institution for the expression of sexuality? That's a whole other question by itself.
Any Hebrew scholars here? The King James says that the sons of God "took wives" from the sons of men, but the Hebrew simply says "they took women". As I understand it, Hebrew does not have a word that specifically means "wife". Rather, a man refers to her as "my woman". (Likewise there is no word for husband, but a woman says "my man".) So I don't think this verse necessarily says that they married these women, just that they had sexual relations with them.
RE Godly line of Seth: There's some justification to say that "sons of God" might mean believers, and therefore perhaps by extension a tribe of believers. But I don't know of any place else in the Bible that uses the phrase "daughters of men" ("bnuth e adam") to refer to sinful people, i.e. that "daughters of men" means people who are not righteous. Perhaps the writer invented the phrase here as a natural contrast to "sons of God". But it's an awfully odd way to put it.
But more problematic, how would intermarriage between "good people" and "bad people" result in a race of heroes? (Whether "nephilim" means "giants" or more like the KJV translation, "mighty men", it still tells us that the result of this intermarriage was notable, famous men.) It makes a lot of sense to say that if angels could interbreed with humans, the result would be something extraordinary. But why would good people and bad people intermarrying result in notable offspring? They're both human, one would think their offspring would be ordinary humans.
IMHO, the angels theory is the more natural reading of the text. Instinctively I find it implausible, but it seems to be what the Bible is saying. The Seth/Cain theory just doesn't really make sense. Or maybe there's some totally different explanation.
My answer is these "Sons of god" were men who had advanced in their level of participation in the organized and authoritative bodies of priesthood governance by making covenants to live their lives in a particular manner. And, evidently, as a part of these covenants, they were expected to only marry women that were sanctioned by the word of the Lord.
An example of this can be seen in the practice of the prophet Nathan to give the word of the Lord as to whom King David should receive as a wife. For David to receive a wife solely according to his own choice would in the eyes of the Lord be considered a sinful act that would put him under condemnation and in great danger. Such was also the downfall of David's son Solomon who took many wives and concubines that were not sanctioned by the word of the Lord through the Lord's Anointed Prophet at the time.
Things are much easier to understand when you take the identities spoken of as Adam, Cain, Abel, Seth, etc. as the collective bodies of priesthood that individual men could become members of by covenant. So, all of the "Sons of God" or the "Sons of Eloheim" are in fact members of these societal bodies that had their own collective identity. Their covenant into a priesthood body makes them all in union and therefore collectively "one flesh". They together form a single body, an Eloheim.
Likewise, Adam is a host of men enlightened to the state of the fulness of priesthood authority in order to receive the dominion, glory, etc. of God in God's own image and likeness. Unfortunately, Adam became beguiled through the counsel of Eve (the Bride or the Church) who was beguiled by the adversary.
Looking at things this way enables you to put aside all of the strange and mysterious alien encounters or non-physical beings. These are normal people who are functioning at a high level of accountability who bring severe cursing upon their posterity for violating their sacred oaths and covenants.
This becomes more clear if you read the Book of Enoch and learn that these men were the "watchers". These are who are referred to in the Book of Jude and in 2 Peter chapter 2. They were regular men who were under covenant in the ancient times who conspired to act sinfully and reaped the consequences.