"Call me Ishmael" -- but don't call me late for dinner.
This is a novel by the science fiction writer Philip José Farmer. It was first published in 1971.
The premise is that Moby Dick ends as it always does, and the character Ishmael is rescued by the Rachel right on schedule. Events then turn weird: A few days later, the Rachel is beset by a storm, and transported a billion or so years into the future. The moon has settled into a lower orbit and appears four times its old apparent size; the sun is large and reddish (that's how science fiction writers let you know that a really long time has passed: They move a handy star an appreciable distance along the main sequence). With the peculiar logic of SF time travel, the ship remains in the same position relative to the Earth without regard to the fact that the Earth itself has moved god knows how many billions of miles in the mean time.
In the future, the seas have dried up, so the Rachel appears in mid-air, miles above the dry ancient sea floor. Again, Ishmael is the only survivor. This time, instead of clinging to a coffin, he lands on a convenient whale. Yes, there are whales in this strange future, flying whales. They're enormous, diffuse creatures much like animate Zeppelins. Most of their volume is occupied by gas-filled bladders. Naturally, there are humans who hunt them from diaphanous airships much like the whales themselves. These people have technology about like what the West had in the nineteenth century, except that they're able to make their sailing ships lighter than air.
It's not many of us nowadays who can get through forty years of adulthood with the same skill set and stay gainfully employed, but our man lucks out. Ishmael learns the language, makes friends, and goes right on hunting whales. Like most fictional time travellers he finds many opportunities for professional advancement, and soon enough he has his own command.
And so on; "many events ensue". There is much danger and strife, yet our plucky hero wins through to find true love in the end. Farmer tries to write a bit like Melville and that's a drag, but you can get through it. It's a shame he couldn't have done something more memorable with such a wonderful premise, but Farmer was like that. He also made a habit of recycling other people's characters and/or historical figures; remember the Riverworld series and all those additions to the Tarzan franchise?