NOT by carbon dating. Old fossils do not contain enough carbon, even if C14 dating could be used.
There are around 40 different radiometric dating methods that have been used. About 12 or these are in fairly common use and include U235 and U238 to two different isotopes of lead, thorium-lead, potassium-argon, caesium-rubidium and other radiometric systems not involving radioactivity, but which can estimate how long a mineral has been in the dark (or in the light). These are mostly used on artifacts so are not used much on fossils. With around 40 methods available, a paleontologist wanting an absolute date will nominate the most suitable for his specimens,
Another method revived in the late 1940s and which threw even more doubt on the Piltdown cranium and jaw is the fluorine absorption test, which measures the amount of F in the fossil and in the surrounding rock or soil. A steady migration of F into the fossils is assumed, depending on the average moisture content and temperature of the soil. This gives a handle on maximum and minimum ages of a fossil and is cheaper than radiometric methods, but probably not by much and would not be as accurate.
Radiometric dates are absolute but expensive to get. Other methods using other fossils for comparison which may require a lens or microscope to see are often used as they are far cheaper and quicker than radiometric dates. The other fossils generally have an absolute radiometric date. Exactly the same methods are considered sufficiently good for most purposes by oil amd mining company exploration geologists with millions of dollars at stake.
If funds are available and there is any informed and serious controversy, a radiometric date may be obtained. Creationist are not considered to be informed or serious controverters and are usually ignored, which is far better than their leaders deserve.
So yes, I know how fossils are dated, which is more than I can say for quite a lot of other people.
.EDIT
"And why a 10,000,000 year old diamond can still measure carbon emissions" What does that mean? I was not aware that a diamond or even a quartz grain could measure anything. Do you mean it still has a C-14 content? Well -citation needed, and apologetics or creationist sites are not citations - let's see the original scientific papers in the original journal by the original author. Otherwise it didn't happen. Far too many lies about C14 over the last 30 or more years for any statement about it to be taken as fact.