What @Harp said. The population bottleneck would lead to genetic drift, which is basically an accident of probability wherein traits get passed on to offspring purely due to random chance, whether they’re beneficial to the species or not. (For instance, if something has a 50% chance of being transmitted, in every new generation it probably won’t be passed on exactly 50% of the time—it might show up in 54% of the next generation. So you might suddenly see a trait increase in frequency even though it confers no adaptive advantage.)
In large populations, natural selection acts against this by killing off the offspring who received maladaptive traits and not killing off the ones who got good traits. In a population bottleneck, however, the effects of random chance (on survival and subsequent DNA transmission) are much more pronounced because arbitrary transmission of a given trait isn’t diluted over a large population, and the effects of chance and luck—rather than fitness—become much more important in regard to what DNA gets passed on.
Over time, whatever DNA was passed on becomes “fixed” in the population—that is, every member has the same alleles at a particular locus. If this DNA was selected for by adaptive pressures, this tends to be a good thing and increases the overall fitness of the population. If the DNA was selected for by accident—which is what happens in genetic drift—it tends to be bad news bears for the population.
One reason this is bad is just because lousy traits have become more common in the population. Another reason this is bad is inbreeding depression. If you have studied a little bit of genetics, you know that people have pairs of alleles, and each of these alleles can be dominant (B) or recessive (b). You get one from each parent. If you get two copies of B (BB), the dominant trait is expressed. If you get B and B (Bb), the dominant trait is still (usually) expressed. You would only express the recessive trait if you get b from both parents (bb).
Sometimes maladaptive traits can survive in a population for a very long time simply because they’re recessive—that is, the gene is carried, but not expressed until you get two copies of it (bb). So if you have a population with increasingly similar DNA (due to drift), the population will start to get more and more copies of this harmful recessive allele, and it will be expressed more and more often, and the recessive disease will increase in frequency even though it’s killing off your population.
And that is why the Noah project would suck.