Of course, most translations say it was a "great fish" and not a whale, but the imagery is seared in many people's minds of that of a whale swallowing Jonah and him being regurgitated 3 days later near Niniveh.
While perusing today's edition of Coastzone Newsletter, they highlighted a site that contains "7 of the Weirdest Human Enigmas". One account mentioned is this one that some of you may not be familiar with....
"The Bible records the account of Jonah who was swallowed by a whale or a great fish, but was later freed from the beast. In 1891, a British sailor lived through the same fate. The seamen aboard the whaling ship The Star of the East struggled to kill a great sperm whale and bring it on board. In the battle between man and beast, two sailors disappeared. But when the whale's stomach and liver were hoisted onto the deck of the ship, it was noticed that something was moving inside the stomach. Cutting open the stomach, the crew found James Bartley, one of the missing men, curled up, unconscious, but still alive."
So it would appear that such an occurance is at least theoretically possible. However if one is of the mindset that miracles do not occur, then it becomes a bit tougher, ahem, to swallow. However if God created the universe, then who is to say that He didn't construct a specific type of fish just for the purpose of landing a scared and disobedient Jonah near where God wanted him?