There you are, reading the book of Jonah as an adult. You know the basic premise. Prophet sent to Ninevah. Decides to run away. Get's thrown in the ocean and swallowed by a giant fish and then vomited out onto dry land...
Wait, what's that? Swallowed by a giant fish and vomited out three days later?
If that random little detail dropped in the book of Jonah doesn't make you pause and ask, "really?", you've probably been in church too long. You've lost your critical edge; a critical edge, I'd argue, that you actually need if you are going to read the Bible correctly and believe what it says. How can you defend the rest of the text in Jonah if you haven't thought critically about this particular detail that is otherwise incredibly difficult to believe?
It's not a sin to be critical of difficult-to-believe things in the Scripture. Even Martin Luther, the theologian and church reformer, said that this was the most difficult of Old Testament texts to believe, and if it weren't in the Bible, he wouldn't believe it. Bringing our critiques to the Scripture allows us to probe more deeply into the nature of what God is doing and therefore bring us to a deeper belief, if we are willing to consider what the text is teaching.
So did Jonah get swallowed by a giant fish or not?
If we're going to ask the question critically, and be true to the Scripture overall, we have to start with an overarching biblical premise, and one of the themes of Jonah: God can do whatever he wants.
That's not an excuse to come up with a non-answer. It's simply the recognition that the God of the Bible is consistently described as the God who created the world and sustains it; who is the God of "the Land and the Sea", as Jonah has said, and therefore, by virtue also the fish of the sea. This God of the Bible can do whatever he pleases within his creation.
If we come to the text critically because we want to disprove God or make him unnecessary, we will not be treating the text fairly. The text assumes the existence of an all-powerful God. It is easy to doubt the story if we don't even want to acknowledge that type of God. If that's the position you are in, that's okay! But you really need to answer that question first–can you accept the premise that there is an all-powerful, sovereign God?–before asking a question of a text that assumes the answer to that question is "yes".
Therefore, the point of our inquiry into the text isn't so much to prove or disprove it, but to discover how God might have worked through his creation. Perhaps we desire to discover the extent of the miraculous; perhaps we simply want to make the story more reasonable to our rational mind.
So, we ask again–understanding that God can do whatever he wants–did Jonah really get swallowed by a giant fish?
Option 1 - God appointed a particular fish for this particular occasion.
If we have come to the text with the premise of the author, Jonah–namely, that God can do whatever he wants–then we must also accept the possibility of the purely miraculous. That is, God created a specific fish for this specific purpose. The answer to how God did this would therefore be unanswerable to us. The details given by Jonah are all the details we need, because they really are all he knew: he was swallowed by a giant fish. The end.
Option 2 - It was an extinct creature that we have yet to discover
The second option which moves a but further away from the purely miraculous is the idea that the fish that swallowed Jonah was an ancient sea creature that we have yet to discover. In the New Testament, Jesus references the story of Jonah and refers to the fish by the same word that is used in other places in antiquity for a type of sea monster that is depicted in drawings as a serpent-like animal that looks exactly like what you would ASSUME a sea monster looks like. The Old Testament books of Job and Psalms both reference a large creature called Leviathon, without going into further detail. Maybe there really was an ancient sea creature called Leviathon that we simply haven't discovered yet.
Archeologists and Paleontologists continue to uncover new species of dinosaurs that were previously unknown. As recently as 2011, a miner in Canada found the petrified carcass of a dinosaur buried far underground! The ocean is a large and vast place; the ocean floor even more so. It's entirely possible that there was an ancient creature that we simply have not discovered as of yet.
Option 3 - It was a Sperm Whale or Great White Shark
Both of thee "fish" would be capable of swallowing a man–particularly a slightly smaller man–whole. If it was a Sperm Whale, which ingests it's food without chewing, Jonah surely could have wiggled down into the belly. A Great White, on the other hand, could swallow a man, and the cold water and slow metabolism of the animal would have kept digestion to a minimum over a three day period.
Of course, none of these explanations do away with the miraculous. In case you are picturing Pinochio's dad, Gepetto, floating around on his boat in the internal caverns of a whale's stomach, or even the Veggie Tale's depiction of Jonah where there is a similar expanse for Jonah to comfortably relax in, you should know that not even a giant Sperm Whale has three days of oxygen in it's belly.
Jonah's point isn't to detail the logistics of his ordeal. He's content to leave it as is, with the understanding that God can do whatever he wants. Jonah's point in this text is to let us know that however it happened, and whatever creature did it, Jonah was rescued by the miraculous intervention of God.
That's what Jonah wants us to know. The God Jonah believed in, and the God Jonah served, is the kind of God who could "appoint" a great fish to come and swallow Jonah, and then "speak to" that same fish to vomit Jonah out on to the store.
He was the kind of God who would pursue a rebellious prophet, even to the depths of the sea.
Further Reading:
http://www.icr.org/article/what-really-swallowed-jonah/
http://www.jewishsightseeing.com/usa/california/san_diego/general_stories/sd9-13-02jonah.html
https://answersingenesis.org/bible-characters/jonah-and-the-great-fish/