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: No one disagrees that the Earth collected from stellar debris. Any Junior High science class explains the conversion of potential energy (height and mass) to kinetic energy (ultimately heat). All those chunks of stuff collecting down into the gravity well produced a lot of heat. By the 19th century, mining exploration had turned up the fact that the solid crust of the Earth had a finite and calculable thickness, and that the interior had to be molten.

How long would it take for the Earth to skim over to a depth where we couldn’t feel the internal heat until we were deep in a mine? The answer that was worked out came to at least several hundreds of thousands of years. This was a century before the internal nuclear furnace in our planet was discovered, pushing the age using this method up to the billions.

We can ignore the astronomical answers, since more and more YEC’s now state that the age of the universe is not a reflection of the age of the Earth; returning to the pre-Newtonian idea that what happens in heaven, stays in heaven; that the Earth is a completely separate entity from anything Up There. But given what was known about the speed of light and the parsec distances of stellar objects, the universe was shaping up to be at least 10’s of thousands of years old. The problem they had in the 19th century was that all the evidence about the age of the Earth was indicating an Earth older than the Heavens! This didn’t resolve until the early 20th century, when astronomers became able to measure galactic distances in the millions of light years and beyond.

Read some of the Young Earth rebuttals, and try not to laugh too hard at the flat statements about what “everybody knows” that have nothing to do with the issue they are talking about.
The uses of innumeracy are classic: Postulating constant rates for things that are demonstrably inconstant (like population growth or orbital decay), waving hands at big numbers to intentionally confuse orders of magnitude (millions, trillions, what’s the diff?), and stating precise numerical conclusions without mentioning the assumptions or the formulas used to get them, allowing no checking, just accepting.

I chuckled when I read the Creationist hourglass analogy for radioactive decay. It stated that we have no idea what the isotope ratios were at the beginning (in some cases we only know to within a few percent, others to parts per million), cannot tell whether isotopes got into or out of the samples in the interim (does anyone have a theory allowing the addition of incompatible atoms into the center of a crystal matrix without isotopic decay?), and that the decay rate might be affected by such factors as heat and humidity (LOL)! Young Earth supporters are fond of using discredited techniques of radiological dating to “prove” that things are younger than the acceptable techniques indicate. They don’t get answers young enough to support the age they are going for, but only to try to show that there is not one exact answer.

They try to apply the legal principle of “Reasonable Doubt” to science. Science uses a different paradigm to determine truth: Rigorous Mathematical Modeling. The concept of error bars is basic to measurement. In 1900, a measurement of anything to within a few parts per million was an ultimate goal. Now, we are measuring things to parts-per-trillion (millionths of a millionth).

Finally, there is the rejoinder that God created the world and universe with billions of apparently ancient clues to mislead and distract people from His Truth. In other words, God lied by providing uncountable bits of evidence to lead astray anyone who will observe His works directly, rather than lying by inspiring men to write a book inaccurate in details unknowable to men at that time.