It is certainly possible that all races we see today trace back to on family that repopulated the earth, starting at around the arguable biblical date of 3000-4000 years ago. However, there are many objections to this claim, each of which must be examined individually. Objections like:
1. Could all of the races come from one family?
You can check out this defense of the biblical position - Where Did all the Races Come from?
In that article, they claim that the differences between the races are not that great, and Noah's family did not just include his own descendants, but his son's wives, who were from other families.
2. Does genetic ancestry research back up such a claim?
In Roots of Human Family Tree Are Remarkably Shallow, the author discusses how modern genetic statistics support the idea that we are quite possibly all related through one modern ancestor about 5000 years ago.
Allowing very little migration, Rohde's simulation produced a date of about 5,000 B.C. for humanity's most recent common ancestor. Assuming a higher, but still realistic, migration rate produced a shockingly recent date of around 1 A.D.
Of course, a Stanford University study indicates that our common ancestor is farther back (70,000 years), but the idea of a common ancestor is certainly supported by science.
3. Other things
You may be interested to read Noah and Human Etymology, which disucsses the similarities in language, esp. how Noah's name was carried across cultures.