Just to add some further discussion to @Pont and @fre0n excellent answers.
The problem of the water needed to submerge the world during the Genesis flood have been discussed for centuries. The narrative could seem legit as traces of marine condition can be seen even in high mountains. The Biblical flood was the most logic explanation for marine fossils and wave patterns far away from any coast. Moreover, at the time when the Bible was written in the Levant, there might still have been a collective memory of sudden sea level rise from the possible opening of Bosporus and flooding of the Black Sea. Other experiences, as tsunamis formed in the tectonic active Eastern Mediterranean would also add to the idea of flooding as a divine respond to human actions.
Early scientists that described and compared features in the nature, e.g. Ibn-Sinaa, could also see that some rocks was formed by water. They added the superposition principle, that younger rocks are situated above older rocks and that the processes that formed the rocks continues today and must have taken long time. What they could see, describe and test was a continuation of the earlier ideas of flooding, but they used reason instead of dogma. Modern disciplines of science, as radiometric dating and taxonomy, have further refined our understanding of time and the relation between water and dry ground, but we are still building on the traditions of earlier scholars as Ibn-Sinaa. Sometimes we prove them wrong, just like future scientists will (hopefully) prove us wrong, but the human wish to understand the nature around us can easily be traced back to Old Testament and before.
Sea levels are constantly changing during the history of Earth. There is a number of parameters that controls the global sea level, some are slow but have a large impact. E.g. during Ordovician, the sea level might have been 3-400m higher than today. Not enough to reach the foot of modern Mt Ararat, but the planet was certainly more blue when the first fishes appeared. Also during Cretaceous, the sea level was 2-300m higher than today and more recent, in early Holocene, the sea level was 120m lower than today. Our ancestors could walk (and maybe a short swim or paddle) from Asia to Alaska, from Java to Australia, and across the Black Sea.
So where did the water go? At colder conditions water can be be trapped in ice shields, glaciers. Antarctica and Greenland holds a rather large fraction of the global fresh water. But most water are in the oceans, and larger sea level changes must be caused by changes of the oceans.
The oceanic crust is, relatively to the continental crust, thin and soft, but get harder and chemically heavier with age. The oceanic crust press down the asthenosphere and this results in more space for water, and the sea level sinks (in relation to the geoid). Mid-ocean volcanism on the other hand lifts the seafloor and forces the sea level to rise. Temperature of water is also a factor to consider. Warm water is has less density and occupies larger volume.
The total change in sea level in the past 500 million years have been estimated to about 500m, this is impressive, but far more than the 3,896m of Mt Ararat, that was supposed to be covered in the Genesis flood narrative. As mentioned, there might be more water trapped in the mantle transition zone, but it's hard to see how it could have slipped out to form liquid water, and even more difficult, how it could get back to the mantle, against gravity, after the flood.
However, as a homage to the Biblical believes, there could be an interesting link between sea level changes and mass extinction. Changing sea levels, especially when it happens rapidly, results in new conditions that species might not have time to adapt to. The connection is discussed in this article by Hallam and Wignall, 1999 (Mass extinctions and sea-level changes). The article is also a good example of scientific methods and reason. The language in a scientific article might not be so colorful as in Genesis, but it can open a world far more complex and wonderful than any dogma.
Finally, a reading tip:
One solution is proposed in Umberto Eco's The Island of the Day Before, maybe the water was borrowed from another day, as the Earth contains two days, separated by the date line.