Radiocarbon dating is an early form of what’s now known as radiometric dating. As noted in other replies, carbon dating is only good for a relatively short span of time. But once scientists figured out the basic idea, they discovered other isotopes with different, much longer half-lives. As noted the core principal is the same: some natural process causes the accumulation of a radioactive isotope at a known rate that stops when whatever it is dies or gets buried. The isotopes then decay, also at a known rate. Measuring the ratio gives you a date that has (ideally) a precision that makes it useful for the time span you’re working with. For example, a +/- 5000 year margin of error would make dating Egyptian pottery pretty useless, but a +/- year margin of 50,000 years would be quite excellent for dating things from the Mesozoic era or earlier.
But there are other ways, most worked out before radiometrics were developed. The most important is stratigraphy. It works on the principal that depth = age. If you have a date for one layer of your dig (due to, say, pottery that was described by a 16th century explorer) and a different date for a much lower layer (due to an example being carved on the base of a sacrificial altar made with wood beams you can date by counting the tree rings), then anything you find between those two layers will be older than the former, and younger than the latter.
You probably already see how problematic this method is. You only get relative dates, not absolute ones. This is why it took the development of radiometrics before it was possible to *prove* things like the age of the Earth, or how long ago the dinosaurs were extinct. But it was, and is, still quite useful. It’s certainly simpler and cheaper.
A combined approach is the most common way: Radiometrics provide exact dates of specific creatures or artifacts. If you stumble across a site that includes these items, it allows you to quickly narrow down how old anything unknown at the site is.
None of this is consistent, simple, or foolproof. Contamination can throw off a dating sample. Plowing, digging, or other sorts of excavations can push newer artifacts into older deposits. The Earth’s crust has been caught in the taffee-pulling machine that drives continental drift for its entire history, making some stratigraphies quite confusing. The machine testing the sample can break, or be set up with the wrong set of assumptions. We’re constantly refining the techniques, which has led to significant revisions of all sort of dates in ways both large and small.
But, overall, the combination of the methods described here and in other answers has provided us with dating tools with a precision undreamt of as recently as a few decades ago. It’s really neat stuff!
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