Genesis 6:1-4

When man began to multiply on the face of the land and daughters were born to them, the sons of God saw that the daughters of man were attractive. And they took as their wives any they chose. Then the Lord said, “My Spirit shall not abide in [or "contend with"] man forever, for he is flesh: his days shall be 120 years.” 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 6:1-4, ESV

The Hebrew phrase rendered here "Sons of God" (ben-'elohiym) is also used in Deuteronomy to refer to Israel and in Job and Psalms to refer to angels. Gesenius's lexicon points out that in the Hebrew language, ben (the son[s] of) can refer not only to physical generation, but also to an association with something (e.g. "sons of wickedness", "son of suffering"). Thus, this passage could be referring to the godly line of Seth's descendants (as laid out immediately before in Genesis 5), or it could be referring to angels (presumably fallen angels). The "daughters of man" are, obviously, human women. Interestingly enough, the word translated man here is the exact same word as is transliterated "Adam". So it is not beyond the realm of possibility that the contrast between the "sons of God" and the "daughters of man" is a contrast between those who followed God and those who, like Adam, rebelled against Him.

Of course, hermeneutically speaking, it is equally possible that this passage refers to the seduction of human women by (fallen) angels. However, this interpretation is doomed to dismal failure theologically, on three key points.

1. Angels do not marry.

Jesus tells us (Matthew 22:30, Mark 12:25) that after the resurrection, people "neither marry nor are given in marriage, but are like the angels in heaven." This strongly implies that angels neither marry nor are given in marriage. The objection may be raised that, perhaps, these antediluvian fallen angels procreated with, but did not ceremonially wed the "daughters of men" (after all, the word here translated "wives" literally means "women" — the Hebrew phrase laqach 'ishshah usually means "to take a woman to be a wife", but is also used to describe Shechem's extramarital relations with Dinah [Genesis 34:2]). However, the mere idea of angelic procreation with humans is problematic in itself, as will soon be pointed out.

2. Angels do not have physical bodies.

We see, multiple times in Scripture, that angels appear or disappear at will (Exodus 3:2; Judges 6:12, 13:3; Luke 1:11, 2:9, 13, 22:43), can sometimes seen by some people but not others (Numbers 22:22-31, II Kings 6:17), and, most tellingly, do not interact physically with the physical world. Jesus' strongest assurance to His disciples that He was flesh and blood, not spirit, was eating (Luke 24:36-43). But nowhere in the Bible are we told of angels eating (earthly) food. In fact, we are given two distinct accounts of food being offered to an angel and the angel refusing to eat it (Judges 6:19-21, 13:15-20).
Someone might bring up Genesis 18-19 in an attempt to contradict this statement. However, I contend that the two men commonly referred to as "angels" may not have been angels at all, but rather prophets. The Hebrew word mal'ak literally means "messenger" and is used in Haggai 1:13 and Malachi 3:1 to refer to prophets. Furthermore, in Genesis 19, the words mal'ak and 'enowshe (which is a generic word referring to human men) are used interchangeably to refer to these mysterious figures.

3. Semidemonic Nephilim raise a number of insoluble problems.

The exact definition of the Hebrew word nĕphiyl is uncertain and can be understood a number of ways; in fact, Gesenius notes that a variation of the word in Chaldean refers to the constellation of Orion. (Gesenius goes on to note that he prefers the interpretation, "fallers, rebels, apostates". However, his wording indicates that this is merely his personal preference and that the actual definition of the word is uncertain.) We see this word used only one other time in Scripture: in Numbers 13:33, when the Israelite spies describe the inhabitants of Caanan; here it is used to emphasize the size and strength of their enemies.

One understanding of the Nephilim, as explained to me by a friend, proposes that the Nephilim were half-human, half-demon hybrids, and could not be allowed to exist (hence the flood). Somehow they repopulated after the flood, giving rise to the Israelites' requirement to leave nothing living of the peoples they drove out of Caanan — no men, no women, no children, not even animals. But aside from the lack of ground for this interpretation of the word and the failure of Scripture to support a semidemonic origin for any species, this explanation contains a further problem: The Israelites were ordered to eliminate every living creature in the Promised Land ... but they failed. (Judges 1 contains a long list of their failures, and Joshua 9:24-27 indicates another. Note also that Rahab, a prostitute and an inhabitant of Jericho which was utterly obliterated for centuries, was not only spared with her family, but also became one of Jesus' ancestors.) Now it is conceivable that God ordained for Israel to spare only the few people who were not part demon, but holding to this understanding of the text requires an unwieldy stretch of the imagination. Is it not more likely that the Bible means exactly what it says — that God had Israel obliterate their forerunners because of their forerunners' wickedness and to demonstrate to the rest of the world that HE IS A GOD BEFORE WHOM NO EVIL CAN STAND?