There are many people who don't believe in it, primarily because its accuracy cannot possibly be verified. Think about it: in the big scheme of things, this dating technology is brand new, yet its ability to determine that something is millions of years old goes unquestioned in the scientific community at large.
But it doesn't. Carbon dating goes back 50,000 years maximum, the relatively short half-life of C14 doesn't allow dates further back than that.
There are two ways to do this. We can date things for which historians know a "right answer". And, we can date things that have been dated by some other method.
Historians don't have "right answers" for really old things. However, carbon dating has done well on young material like the Dead Sea Scrolls, the Minoan ruins, and acacia wood from the tomb of the pharoah Zoser.
Some corals can be carbon dated, and also dated by another radioactive material, Thorium-230. Pollen found in the Greenland icecap has been carbon dated, and also dated by counting ice layers. The three methods confirm each other.
Trees grow a thick ring in a good year, and grow a thin ring in a bad year. It is sometimes possible to match up tree-ring patterns between different trees. When enough suitable trees are found, living or dead, the matching is completely accurate. Then, we have wood for which we know the right answer.
So, carbon dating has been calibrated against the rings of California bristlecone pines, and Irish bog oaks, and the like. When this was first done, it turned out that carbon dating had been giving too-young dates for early civilizations. Apparently, the production of C14 by the Sun has changed by several percent across the last 10,000 years. We know (from other measurements) that the Sun hasn't fluctuated by more than 10 percent in the last million years. However, even this small an adjustment was a bit of a shock. For example, Stonehenge suddenly became older than the Pyramids, instead of younger.
Since then, several other calibrations have been done, which confirm and extend the tree-ring one. Some were done by finding lakes with atmospherically derived carbon in their annual layers of silt (called varves). In those particular lakes, the varves can be counted, and the varves can also be carbon dated. See below for details about the 45,000 annual varves in Lake Suigetsu.
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Are There Inaccurate Carbon Dates?
Yes. There are three kinds.
The first kind are datings of things that should't be carbon dated. For example, polar bears that eat seals aren't getting their carbon from an atmospheric source.
The second kind are datings on contaminated samples, or on samples which are a mixture. Old samples contain much less C14, so the measured date of older samples is strongly affected by even small amounts of contamination.
The third kind are dates which were measured before the 1970's. In the 70's:
* much better measurement equipment was introduced. the tree-ring calibration eliminated the assumption about the Sun being constant.
* procedures for avoiding and recognizing contamination were established.
In short, all carbon datings published in the 1950's and 1960's are suspect.
Right. Carbon 14 dating is only good out to 50,000 years or so. Due to the halflife, don't you know?