1 The LORD then said to Noah, "Go into the ark, you and your whole family, because I have found you righteous in this generation. 2 Take with you seven of every kind of clean animal, a male and its mate, and two of every kind of unclean animal, a male and its mate, 3 and also seven of every kind of bird, male and female, to keep their various kinds alive throughout the earth. 4 Seven days from now I will send rain on the earth for forty days and forty nights, and I will wipe from the face of the earth every living creature I have made."
5 And Noah did all that the LORD commanded him.
6 Noah was six hundred years old when the floodwaters came on the earth. 7 And Noah and his sons and his wife and his sons' wives entered the ark to escape the waters of the flood. 8 Pairs of clean and unclean animals, of birds and of all creatures that move along the ground, 9 male and female, came to Noah and entered the ark, as God had commanded Noah. 10 And after the seven days the floodwaters came on the earth.
11 In the six hundredth year of Noah's life, on the seventeenth day of the second month—on that day all the springs of the great deep burst forth, and the floodgates of the heavens were opened. 12 And rain fell on the earth forty days and forty nights.
13 On that very day Noah and his sons, Shem, Ham and Japheth, together with his wife and the wives of his three sons, entered the ark. 14 They had with them every wild animal according to its kind, all livestock according to their kinds, every creature that moves along the ground according to its kind and every bird according to its kind, everything with wings. 15 Pairs of all creatures that have the breath of life in them came to Noah and entered the ark. 16 The animals going in were male and female of every living thing, as God had commanded Noah. Then the LORD shut him in.
17 For forty days the flood kept coming on the earth, and as the waters increased they lifted the ark high above the earth. 18 The waters rose and increased greatly on the earth, and the ark floated on the surface of the water. 19 They rose greatly on the earth, and all the high mountains under the entire heavens were covered. 20 The waters rose and covered the mountains to a depth of more than twenty feet. 21 Every living thing that moved on the earth perished—birds, livestock, wild animals, all the creatures that swarm over the earth, and all mankind. 22 Everything on dry land that had the breath of life in its nostrils died. 23 Every living thing on the face of the earth was wiped out; men and animals and the creatures that move along the ground and the birds of the air were wiped from the earth. Only Noah was left, and those with him in the ark.
24 The waters flooded the earth for a hundred and fifty days.
Yeah, yeah, I know, it's a miracle. It doesn't matter how physically preposterous it is, God can do anything. Sure, it appears that this could only have been written at a time when people had no idea of the true nature of the earth, but hey, it's God, reality is irrelevant. There's no sense my going into the water cycle -- rain cannot cause the seas to rise because the source of rainwater is evaporation from the oceans -- or the height of the mountains, or any of that. The believers will believe, and that's all there is to it.
It's much more interesting to consider the near ubiquity of flood myths in cultures from around the world. Some of these clearly could have arisen by diffusion from the same ancient culture that gave us the myth of Noah. For example, the Greeks had a parallel myth about a fellow named Deucalion who built an ark and took his (multiple) wives and animals on it to ride out a great flood. The Romans also had a version of this myth. The Masai of East Africa have a strikingly similar myth as well, although it could easily be derived from contact with Christians in historical times, but there are similar myths from remote areas of Asia that seem less likely to have derived from modern contacts. Other flood myths aren't so similar -- they lack the ark, and have additional elements such as dragons and so forth. The lucky people who are forewarned may survive by fleeing to a mountaintop, or a few people survive because they happen to live on high ground already. In other versions, there are several people or families who are saved on boats or various floating objects. The boats may be anchored, or in one case the ark was destroyed by the devil before the flood came, so God was forced to provide an iron ship for the man he chose to save. In the Hindu version, Manu was saved from the flood by a fish. In a Chinese version, two children survive by floating in a giant gourd.
In my quick review, myths from the Americas are highly dissimilar to the Noah story, whereas tales from various places in Europe, Asia and Africa are notably similar. I'm no expert but this suggests to me that the Noah-style stories come from a common source, rather than some psycho-dynamical archetype as some have proposed. Such a dramatic story could certainly have diffused widely, and if you look at the distribution of myths with floods and arks, it isn't far fetched to imagine an origin in central or western Asia.
One popular candidate is the hypothesized creation of the Black Sea by the Mediterranean overflowing what was then a mountain pass and is now the Bosporus in about 5,600 BC, resulting in the innundation of more than 100,000 square kilometers within a few weeks. This theory is controversial, but if it happened, it must have made one hell of an impression on people in the area - who would, of course, have been dispersed widely thereafter and taken the memory with them. Some would have fled on foot, and some indeed might have saved themselves by jumping into boats. (Christians criticize this hypothesis on the grounds that the Black Sea flood would not have been very similar to Noah's flood, but that obviously doesn't carry any weight with me.) Later studies have cast doubt on the hypothesized 5,600 BC event, but support an earlier and rather different flood in which the Caspian overflowed into the Black Sea.
All very intriguing. Perhaps we'll have a definitive answer one day. Meanwhile I'll just say that all preliterate peoples were unaware of the extent of the wide world, and generally would have thought that the lands they inhabited constituted a good portion of it. Therefore any major flood would have seemed to them like the drowning of the whole world, and of course would have been a very memorable event, to say the least. Suppose that the people of New Orleans in 2005 were unaware of any lands beyond Plaquemines and the north shore of Lake Ponchartrain. How might they have interpreted the hurricane?