The 19th century hosted a couple of intellectual developments that drew fresh attention to the story of Jonah in the Hebrew Bible. First, the birth of modern biblical criticism from the 18th century had drawn fresh attention to questions about the historicity of Old Testament stories. Although the rift within American Protestantism with the invention of "literal" (spoiler alert: not really, see also Ezekiel 16 and 23) interpretation in order to defend slavery is more well known today, this also became a major sticking point between liberal/modern Protestant scholars and Catholic scholars. Way back in the 16th century, of course, Martin Luther had barred several traditional books from the Old Testament on their lack of provable historicity, but now other books were coming under fire.
In 1792, for example, Moravian (then HRE, now Czech Republic) theologian and philologist Johann Jahn--very Catholic--had published an introductory guide to the Bible. He argued on the basis of literary and linguistic evidence that some of the deuterocanonical books barred by Luther were literary creations for the purpose of teaching nice lessons. Crucially for our purposes, though, he lumped the Book of Jonah (which was and is in Protestant Bibles) in with them. The Catholic Church was...not happy, to say the least. His scholarship was censured and he lost his university chair.
The tensions over Jonah were alive and well as much as a century later. You can hilariously see them play out in the 1911 Catholic Encyclopedia--scroll down to "Historicity." (Hint: it's most of the page.)
Not a single Father has ever been cited in favor of the opinion that Jonah is a fancy-tale and no fact-narrative at all...To the Rationalist and to the advanced Protestant Biblical scholar these arguments are of no worth whatsoever. They find error not only in Jewish and Christian tradition but in Christ Himself. They admit that Christ took the story of Jonah as a fact-narrative, and make answer that Christ erred; He was a child of His time and represents to us the ideas and errors of His time. The arguments of those who accept the inerrancy of Christ and deny the historicity of Jonah are not conclusive.
So there was, on one hand, a whole pile of curious resentment/resentful curiosity of the historicity of the Jonah legend. And on the other was the rise of natural history and naturalism in the public eye.
One of the really awesome things about the explosion of interest in natural history as a more scientific endeavour than earlier "making collections out of everything as curiosities" is that so much of the work was concentrated in museums--putting all the developments, cool findings, animal displays that would NEVER pass ethical muster today, right in the public eye. (Plaster casting was gaining ground as its practitioners gained skill, though--in fact, lifelike whale models at the Smithsonian and the American Museum of Natural History were huge national news in the 1900s). And there was a massive market for it, too, in terms of public interest.
Of course, one of the popular things to display--and see--was preserved animals and skeletons. And so you have a trickle of reports from the late 19th and early 20th centuries of, you guessed it, visitors taking the chance to see if they, like Jonah, could fit inside the belly of a whale.
The most (in)famous account, which still gets play in fundamentalist Christian propaganda, is a 1928 recounting of a ca. 1900 youthful escapade:
My own experience was in Birmingham about 25 years ago, when teh carcass of a whale was displayed for a week on vacant land in Navigation Street...I was one of twelve men who went into its mouth, passed through its throat, and moved about in what was equivalent to a fair-sized room. Its throat was large enough to serve as a door. Obviously it would be quite easy for a whale of this kind to swallow a man.
There are a number of similar stories springing up around the time. Not formal experiments, but curious people. Did they actually happen, or is this the sort of "No s---, there we were" story that grows in the telling?
It's interesting to note that there was also, at this time, something of a fad for repeating rumors of modern-day fishermen swallowed by whales. Biblical philologist Paul Haupt in 1907 (who does not accept Jonah as a true story, yet devotes pages upon pages to the question of whether the sea-creature was a sperm whale or a shark) throws the kind of scholarly shade that can only come out of biblical studies:
It is of course impossible that Jonah should have been alive...after three days and three nights, although it is reported in [German newspaper] that the whaleman James Bradley was, in Feb. 1891, swallowed by a whale, and on the following day he was taken alive out of its stomach...The sailors had much difficulty restoring him to consciousness. It was not til after three months that James Bradley recovered his reason. He seems to have been much more fortunate in this respect than some distinguished Biblical scholars.