The exodus story is supported by Egyptian records. They have a different slant on it, of course, but they record the event.
The global flood did occur, though not the way it is popularly understood. Rising sea levels from the end of the last ice age sugmerged several cities and nasceant civilizations that existed before Babylon, Egypt, etc. Submerged cities and structures have been located off the coast of India and Japan. The Black Sea was originally a fresh water lake until sea water from the Mediterrenean came pouring in (itself a large valley until about 80,000 years ago when rising sea water from an even earlier and more massive ice age melt off flooded it). Archeologists long thought that rivers were the cradle of civilization because they permitted farming and trade; now it seems that rivers were where fleeing inhabitants of flooding coastal cities went. It also explains why so many geographically diverse civilizations seemed to spring up within a few centuries of one abother along the great rivers of the world: Coastal civilizations would have been in contact with one another long before that and the rising sea level would have only isolated them by forcing them upriver.
The ark is often depicted in art as a boat, but a more accurate depiction would make it more like a barge. At the time Noah and his family were living in the Black Sea basin; to them it must have very much seemed like the whole world was flooded.
And we have to ask what exactly was meant by "the world" in the story of Noah. Was it literally the entirely planet covered in water? Or was "the world" used metaphorically, the same way Caesar taxed "the world" at the birth of Christ. Did the Meso-Americans and Chinese send tax dollars to Rome?
"The world" to Noah consisted of the Black Sea basin, and there would have been far fewer animals to catch and carry than if the entire planet was involved. Further, assuming Noah wasn't stupid, he probably carried the youngest animals he could find to save space and supplies aboard the ark: A pair of lion cubs would be far easier to handle than a full grown pair of lions, right?