After Yom Kippur I did a little Googling. It was easy to track down the story I remembered reading - and to track down its debunking. The story as I remembered reading in one of its earlier versions:
The whaling ship Star of the East, was in the vicinity of the Falkland Islands, searching for whales, which were very scarce. One morning the lookout sighted a whale about three miles away on the starboard quarter. Two boats were manned. In a short time one of the boats was near enough to enable the harpooner to send a spear into the whale, which proved to be an exceedingly large one. With the shaft in his side, the animal sounded and then sped away, dragging the boat after him with terrible speed. He swam
straight away about five miles, when he turned and came back almost directly towards the spot where he had been harpooned. The second boat waited for him, and when but a short distance from it he rose to the surface. As soon as his back showed above the surface of the water the harpooner in the second boat drove another spear into him. The pain apparently crazed the whale, for it threshed about fearfully, and it was feared that the boats would be swamped and the crews drowned. Finally the whale swam away, dragging the two boats after him. He went about three miles and sounded or sank, and his whereabouts could not be exactly told. The lines attached to the harpooners were slack, and the harpooners began slowly to draw them in and coil them in the tubes. As soon as they were tauten, the whale arose to the surface and beat about with its tail in the maddest fashion. The boats attempted to get beyond the reach of the animal, which was apparently in its death agonies, and one of them succeeded, but the other was less fortunate. The whale struck it with his nose and upset it. The men were thrown into the water, and before the crew of the other boat could pick them up one man drowned and James Bartley had disappeared. When the whale became quiet from exhaustion the waters were searched for Bartley, but [he] could not be found; and under the impression that he had been struck by the whale's tail and sunk to the bottom, the survivors rowed back to the ship. The whale was dead, and in a few hours the great body was lying by the ship's side, and the men ere busy with axes and spades cutting through the flesh to secure the fat. They worked all day and part of the night. They resumed operations the next forenoon, and were soon down to the stomach, which was to be hoisted to the deck. The workmen were startled while labouring to clear it and to fasten the chain about it to discover something doubled up in it that gave spasmodic signs of life. The vast pouch was hoisted to the deck and cut open, and inside was found the missing sailor, doubled up and unconscious. He was laid out on the deck and treated to a bath of sea-water, which soon revived him, but his mind was not clear, and he was placed in the captain's quarters, where he remained to [sic] weeks a raving lunatic. He was carefully treated by the captain and officers of the ship, and he finally began to get possession of his senses. At the end of the third week he had finally recovered from the shock, and resumed his duties.
The straight dope debunks the story with the following points, building on an article by Edward B. Davis:
(1) The story has appeared over the years in numerous publications, both secular and religious. An 1896 story in the New York Times gives essentially the account above and says it came from "The Mercury of South Yarmouth, England, in October 1891," but it sounds a little skeptical.
(2) The Yarmouth Mercury of August 22, 1891, carried a story entitled "Man in a Whale's Stomach / Rescue of a Modern Jonah," which gives the account above. There's no byline nor any indication that the writer spoke with Bartley, the ship's captain, or any of the sailors.
(3) In June 1891 a rorqual whale was beached near the town of Gorleston, just south of Great Yarmouth, and was killed, stuffed, and exhibited around England.
(4) Sperm whales are capable of swallowing humans. They live on squid, which they swallow whole. In 1955 a 405-pound squid was removed intact from the belly of a sperm whale.
(5) In 1906, Lloyd's of London reported that Star of the East, a British ship, had set sail from Auckland, New Zealand, in December 1890 and arrived in New York in April 1891, so it might have been near the Falkland Islands in February. However, it was not a whaling ship, and there was no James Bartley on the crew list.
(6) Also in 1906, the wife of J.B. Killam, captain of the Star of the East, wrote that she'd been with her husband all the years he commanded the ship and that no one had been lost overboard during that time. "The sailor has told a great sea yarn," she said.
If you have the time I suggest you read this article by The Skeptical Review, which is a much longer but very enjoyable read of how Davis went about researching and ultimately debunking this story. Alternatively I suggest you read the full article by Davis, here.
I should have known it was too much of a whale of a tale..