No, just... no. Evolution does not work that way. Evolution does not generate new species within a span of 6000 years. Or even 200,000 years. Not like that, not for complex higher mammals.
If the Abrahamic flood story is real -- and it will have to compete with the Babylonian flood story -- then it's possible 'two of every animal' meant two of every domesticated animal on his farm/land. That would make more sense.
But there is no evidence whatsoever of *all* humans dating back to any flood. Mitochondrial Eve dates back to at least 200,000 years ago, when homo sapiens sapiens was just developing, and she was in sub-Saharan Africa. If you really want to contemplate that all of humanity arose from a single population, that is the current theory; and the holes in it suggest that the single population originator dates back even FURTHER.
Could Mitochondrial Eve's population of h. sapiens gotten on an ark to escape a flood? Doubtful. They were budding tool users but not to the extent that it would take to build even a coracle boat. Plus, they were not in a region in which there is geological evidence that there was any flooding.
The best contender for the flood myth is the recurring theme of floods in Egyptian mythology. Egyptian history -- history, not myth -- also has some intriguing events to look into, in particular the establishment of Atenism (the first true monotheism) and the civil war in the Late Bronze Age between Upper Egypt and Lower Egypt, which was under the control of the Hyskos and Sea Peoples (called in Egyptian the 'Habiru.') The Hyskos were driven out of Egypt, and one theory is that these people went to Canaan. After a period of integration and assimilation, Atenism reasserted itself and the priests of El' (sound familiar?) declared him to be the one true god, purging the priesthoods of other deities including Baal Hadad and Asherah.
There's some fascinating things in the actual history of the world, before, during, and after the Bronze Age Collapse. Between the Sea Peoples and the detonation of Thera/Santorini, the the disappearance of Minoan civilization, there's a gloriously complex world that we've only just begun to uncover. Sadly, if you take the Abrahamic Bible as anything other than metaphor, mythology, and religious scripture, you are going to be sorely, sadly disappointed.