Utter madness. Do you have any idea how many extant species there are in existence? Two of every reptile and mammal, the majority of which did not, in fact, live within walking distance of Noah's residence? Oh, he'd also have to have cultivated a sufficient number of every single plant species in order to ensure they stuck around. Oh, he'd also have had to have collected viruses and bacteria, even though he had no idea what they were and had no possible way to see and contain them. Oh, yeah, also, all the aquatic life that couldn't survive in saltwater, or adjust properly to the change in depth.
But, no, you're right, that totally could have happened.
Also, just in case you weren't aware, merely two of every animal would only produce viable offspring for one generation, after which the entire process would succumb to inbreeding.