It is absolutely impossible !!
As you said, there is not enough genetic diversity, and there isn't enough water in the world.
But let's pretend that a lot of water just evaporated into space since then... and that there was enough water back then to flood the whole world.
Ok, let's start. There are millions of animal species in the world, so you'd need a boat big enough to carry millions of species times 2, because that's what it said in the story, but since it's impossible to continue a species from one couple, this million species would have to be multiplied by hundreds of couples. So I have just multiplied millions of species by hundreds, which would require a enormous amount of space, and a "humongous" boat, one which would make today's biggest battle ship look like a little toy next to it.
Now lets look at the build and the time it would take.
This boat would be larger than the Egyptian pyramids, which was claimed to be made in 20 years only, but if you look at the size of the pyramids, in order for it to be made in 20 years, they would have had to extract, sculpt, and carry it to the place where the pyramids were every 5 SECONDS, even with the labor of thousands of slaves, so clearly these pyramids were built in a much longer time, I would say at least.. 100-130 years.
So let's make a comparison, this boat would certainly be bigger than the pyramids to fit millions of animals in it, so think, if the pyramids were built in 100-130 years with the help of thousands of slaves... the arch with the labor of only a couple families would take a MUCH longer time to be built, I would say 2000 years... come on I'm being generous..
So what? Did Noah have slaves?
Well, wouldn't that make Noah a bad guy? Which would contradict the reason God chose him to stay alive.. oh wait no, the Biblical God is OK with slavery, sorry.
But how would a couple of families control thousands of slaves?
Was Noah that rich to hire so many slave drivers?
But still, even with the help of slaves that arch would take longer to build than a man could ever live.
And what did these animals eat? There would need to be a huge amount of food for millions of animals to eat in a boat.. so just like that I made the arch much bigger which means, more time for it to be built.
And who fed these millions of animals? How could a couple of families feed millions of animals?
Even thousands of slaves couldn't, and if they had slaves in the boat... what happened to them?
Did noah kill them after the flood? Did he kill the slave drivers too?
WHAT A BLOODSHED!
And one more thing... A structure so big could never be built out of WOOD! It would collapse easily!!
OK so now you and me know it's a lie OK!
Have a good day.