This is more about history of science than about modern answers. Some of the latest methods for calculating how old the Earth is are succinctly summarized here at www.talkorigins.org.

One of the more readable of the many innumerate and unscientific rebuttals to a few of the dating methods can be found at http://www.allaboutcreation.org.

In brief, before the 18th century scholars generally accepted that the Earth was created shortly before man began keeping records. But then came “the Enlightenment”, and systematic record keepers began turning up (correlating) all sorts of things that could only make sense if the world were older than previously assumed.

For example, fossils were already a problem for a Young Earth outlook in the 1700’s. Given the predictably small percentage of any animal population that gets caught in the conditions that allow fossilization (one big flood lasting for weeks won’t result in fossilization), and the number of different fossil species found, there isn’t room on the planet for them all to have had living populations in the same few thousand years! Fortunately, different fossils are found in specific layers, the same layer for a given fossil type anywhere on the planet, allowing them to be spread out over a few hundred different epochs. Of course, this stretches the world time into at least millions of years.

Another example: …