According to the Book of Genesis, Noah saved his family and representatives of every animal from a flood by constructing a boat. So since all of humanity should be descended from Noah, does that mean everyone is Jewish?
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As David Perlman stated in his answer, Biblical Noah was not Jewish. Depending on how you define Judaism, the first "Jew", or rather, the first person to recognize the G-d that Jews worship as the Creator and Master of the universe, was Abraham.
If you are interested in nationality, it goes something like this: Noah and his children were the only survivors of the Biblical flood. This means that everyone alive today stems from them. Nationality, then, as outlined in Genesis, stems from them as well. Noah had three sons: Shem (or Sem in English), Ham, and Yefeth (or Japheth in English). See here for a more extensive, but pretty clear, treatment of this: http://en.wikipedia.org/wiki/Sons_of_Noah
Abraham was a 10th generation descendant from Noah through his son Shem. Jews and Muslims believe they are descended from Abraham via his sons Isaac and Yishmael, respectively. These are the only two groups (that I know of) that identify as Semites, and thus the reason "anti-semitism" has, since the 19th Century, been the name given to hatred of Jews, and the reason Muslims in the latter half of the 20th Century began to object to this term.
So, if you are not descended from Abraham, you are most likely not descended from Shem, but one of the other two sons. Traditionally, Europe is descended from Yefeth and Africa is descended from Ham, but some have viewed this as racist theory because Ham was cursed due to some impropriety that involved his embarrassing his father (Noah) and not treating him with proper respect.
But without getting into all of that, the short answer to your question, based on the additional comments you made to David Perlman's answer, is no, not everyone is Jewish. Only the Jews are Jewish, but we're all cousins.
It's kind of like if Johnson has three kids, and they each have kids, and 9 generations later there's a Johnson daughter who marries a Smith, and they have kids, not everyone who descended from Johnson is automatically a Smith just because one set of Johnson's great-great-great-great-great-great-great-great-great-grandchildren are Smiths.
Biblical Noah was not Jewish. The first Jew was Abraham. Abraham came along ten generations after Noah.
From a Jewish perspective all people are children of Noah. All children of a maternal descent of Abraham, Isaac and Jacob are Jewish.
So no, not everyone is Jewish.
Thanks to Seth's comment I did a bit more research on the topic.
The question of who is a Jew is a halachic question. Halacha states (Even Haezer 44) that Jews are those men and women born to a Jewish mother. This is based on the Tamlud Kidushin 68b.
The keeping of halacha began with the giving of the Torah at Mount Sinai. From that point on Jewish nationality became based on maternal descent.
However, the original question relates to the time of Noah, some four hundred years before the giving of the Torah. To generalize the question we could ask to whom was the Torah given? Who were the people who stood at Mount Sinai and were declared the people of Israel?
I could not find a halachic source for this, and the number of Midrashim that deal with this period seem to contradict one another. There must have been some criteria ... and this seems like a good question to post on it own ...