What exactly is it about scientific dating that is in error?
Every method of dating has assumptions based on an a priori belief that the age of the earth is really really old. If those assumptions are wrong, the dates are wrong. With radiocarbon dating, they assume the earth's production and decay of carbon are in balance which is not really true. Many things dated to be millions of years old can also be explained by a single catastrophic event such as Noah's flood. We've even seen severe and wide-spread tsunamis, hurricanes (Katrina), volcanos, severe electrical storms, floods, droughts, in our life time. What's the chances of everything being nicely in balance for millions, or billions of years?
Methods of dating: In practice, if a radiocarbon date fits the preconceived theory, they include it, if it doesn't, they reject it. There is considerable amount of bias and weight put on those a priori assumptions (eg that the earth really is old). Those assumptions come from evolutionary theory which requires very long ages in order to work in the first place.
In the case of aboriginal rock paintings in Australia, they are dated to be tens of thousands of years old. Based on paintings of animals which are extinct - again, itself based on an assumption that they became extict 10's of thousands of years ago, when that is not really known.
Based on radiocarbon dating - they can't date the pigment itself (need enough organic material in it) but only the rock surrounding it I think. Even so this is the most prone to error method.
These paintings are merely clay and water. They can fade fairly easily yet are supposed to have lasted tens of thousands of years. They have probably been touched up (painted over) from time to time , and even due to water damage they've had to put coatings on them to stop them fading away - now what's the chances that these paintings lasted 30,000 years if they're worried about them fading away in a generation?
In the same areas where they have these paintings supposedly dated thousands of years old, there are aborigional paintings of Europeans with guns and ships which look pretty much the same as these thousand year old paintings.
Scenes of organised warfare in thousand of year old aboriginal paintings dated at 30,000 years contradict the evolutionary view that warfare only began after agriculture developed and resulted in conflict over territorial control of land.
They deliberately ignore other evidence we have such as aboriginal stories or legends that parallel Noah's flood and the tower of babel.
All in all, the evidence doesn't stack up to tens of thousand year histories, at least not conclusively.
The current methods available and the issues associated with using them are explained here:
http://www.une.edu.au/archaeology/WorldRockArt/dating.php
The one about getting enough material veruss ruining the ancient artwork is an interesting one. Those dates amounting to 30,000 years can easily drop back to 10-12000 years or even 6000 years.