I’m not saying that I am on board with their explanation, but here’s an answer from ANSWERS IN GENESIS
http://www.answersingenesis.org/articles/nab3/how-could-animals-fit-on-ark
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How Did Noah Fit All the Animals on the Ark?
To answer this question, we must first ask how many animals were actually on the ark. Critics have fantasized the presence of millions of animals overloading the ark. In actuality, the Bible makes it clear that the cargo was limited to landbreathing vertebrate animals—corresponding to modern birds, mammals, and reptiles, as well as their extinct counterparts.
Was every species on the ark? No! From chapters such as Leviticus 11, it is obvious that the created kind (min in Hebrew, in Genesis 1:11–12, 21, 24–25) was a much broader category than the modern term of classification, species. Current baraminological research suggests that the created kind most closely corresponded to the family level in current taxonomy. However, to be conservative in this study, the genus was set as equivalent to the original created kind. As for the clean animals that entered the ark in seven pairs, this added a modest number of additional animals, notably bovids (cow-like mammals) and cervids (deer-like mammals). Under these conservative assumptions, there were no more than 16,000 land animals and birds on the ark.
According to the Bible, the ark had three decks (floors). It is not difficult to show that there was plenty of room for 16,000 animals, assuming they required approximately the same floor space as animals in typical farm enclosures and laboratories today. The vast majority of the creatures (birds, reptiles, and mammals) are small. The largest animals were probably only a few hundred pounds of body weight.
It is still necessary to take account of the floor spaces required by large animals, such as elephants, giraffes, rhinos, and some dinosaurs. But even these, collectively, do not require a large area. God would likely have sent to Noah young (and therefore small, but not newborn) representatives of these kinds so that they would have a full reproductive potential for life after the Flood to repopulate the earth (Genesis 7:1–3). Even the largest dinosaurs were relatively small when only a few years old.
Without tiering of cages, only 47 percent of the ark floor would have been necessary. What’s more, many could have been housed in groups, which would have further reduced the required space.
What about the provisions for the animals? It can be shown that the food would have filled only 6 to 12 percent of the volume of the ark, and the potable water only an additional 9 percent of the same.
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How does the potable water take up only 20% of the physical space of the animals when most creatures will take in over 100% of their body weight in just a few weeks?