No, this is not naturally possible. All "fish" fill their mouths and stomachs with water. A person in said mouth or stomach would therefore drown very quickly regardless of what animal they were inside of. Furthermore, only sperm whales are physically capable of swallowing a human. The sperm whale's stomach could not hold enough air for a resting man to survive three days even if it were full of air. (A person would need somewhere around 6,000–36,000 liters of air to survive. Sperm whales' first stomach can hold 300 liters. Assuming that their four stomachs are at least close to each other in size, which is reasonable from my research, that's not going to be enough.) And again, sperm whales do not swallow air to begin with.
The Bible does not claim that this is some run-of-the-mill activity that must be explained by naturalistic means however, so this a rather pointless inquiry to begin with. Even when this question was asked on Christianity.SE, they gave the same answer. The book of Jonah has other miracles in it, such as the storm stopping when Jonah is thrown off or Jonah then being vomited up where he was told to go after he repented, and the Bible as a whole obviously recounts many other miracles being done. Taking the Bible literally does not mean that it must be explained naturalistically. With the help of the God of the Bible, this would obviously be possible as He is omnipotent. Without Him however, its obviously not.