The fallacy of Noah's story lies on the absurdity of the writings.
1) Impossible for Noah to built the Ark. Even today we can not built 450 feet wooden ships
2) The ark would be rotten by the time the flood begun.
3) Impossible for all the animals of land and sea to gather at Israel.
4) Impossible for all seeds of plants to gather at Israel.
5) Impossible for all animals including their necessities (e.g. food, waste space, aquaria for marine animals) to fit the ark. An estimated 15000 species are discovered each year which means that in order for them to survive today, they were all saved by Noah!
6) Impossible for Noah to establish the gender and the fertility of each animal. Some animals are hermaphrodite and also others live in colonies only (e.g. bees) .
7) Impossible for all the animals to enter the ark in one day.
8) Impossible for the ark not to sink like a brick the moment it was partially loaded.
9) Impossible for animals to survive inside the ark in the conditions described, for a year.
10) Impossible for such amount of water to exist in order cover Mt. Everest.
11) Impossible for the ark to sail into hurricane waves and also tsunamis caused by constant eruptions.
12) Impossible for 8 persons to feed and clean all the animals. It is estimated that a staff in excess of 50,000 would be needed to care for all creatures aboard.
13) Impossible for marine animals to survive in sea because of destroyed salinity, rock/earth to water ratio and boiling waters.
14) Impossible for the marine animals to survive in aquaria without constant mechanical refreshing of sea water, which actually did not exist.
15) Impossible for all the animals to survive inside the ark due to lack of ventilation.
16) Impossible for any tree, including an olive tree to have survived one year under volcanic, boiling waters.
17) Impossible to redistribute the animals and plants after disembarkation.
18) Impossible to carry the aquaria with all marine animals from Mt. Ararat to all lakes and sea.
19) Impossible for short-lived animals to survive.
20) Impossible for diseases to survive.
21) Impossible for any survivor be it human or animal to remain alive after disembarkation. Eventual starvation would kill everyone.
22) Impossible for the ethnic complexity found throughout the world to be derived from the five only flood survivors in the few centuries since that time. (the sons of Noah do not count).
23) Impossible for all humans to have died except the Noah family. Egyptian records show the dynasties of the pharaohs continued unbroken since 3100 BC. The flood was supposedly happened around 2345 BC.
24) Impossible for Noah to have or obtain such advance technology required for building the Ark, to live for 350 years afterwards and not to pass this technology to future generations. After the Ark, people returned to reed rafts and hollow logs for centuries.
25) Impossible for God to make mistakes as appears from being sorry for creating people.
26) Impossible for God not to be Omniscient and not to know that after few years people would be sinners again, therefore there was no point for the flood.
27) Impossible for God not to be Just and kill all animals and plants for no reason.
Gregg Easterbrook a Christian scientist in his 1997 book "Science and God: a warming trend" said, "Creationism is an incredible pain in the neck, neither honest nor useful, and the people who advocate it have no idea how much damage they are doing to the credibility of belief."
Each of the above main arguments has sub-folders enlisting additional arguments and of course there are arguments concerning theology and morality.
In short, there can be no average intelligence person who can still claim that the Noah's story as described in Genesis can be true. If God wanted to punish humans, he could just send a deadly human disease, instead of having to perform millions of miracles and wait for about 55-75 years for the Ark to be built. This way, he could also save the lives of the innocent animals and plants. But again this fairy tale was written by primitive, ignorant, goat herders.