Answer:
Even though radioactive dating's accuracy is plus or minus a few decades, the accuracy of ancient writings is far worse. Consider the Bible which is full of stories about how mankind originated and developed. If you believe every single word in the Genesis, then the world is less than 10,000 years old.
But since not everyone is Christian or Jewish or Muslim (the old testament is basically the same for all three), lets take ancient Chinese writings where dragons and spirits ruled the world and they lived for hundreds of thousands of years, and humans lived enough to record their ages.
What other ancient texts can we use? Most of them are exaggerations of some real life event that happened and was told from one generation to the next until someone learned how to write and created their own version. The problem with stories that pass from one generation to the next is that they are never repeated or passed 100% exactly like the way they happened.
Also remember that humans back in ancient times lived around 30 years only, so a 200 year old story would have been told many times from a 30 or so grandparent to their children and then to the grandchildren.