You do not date 68 million year old fossils with radiocarbon dating, as the upper limits of that method are in the 50,000 year range.
The method also is not inconsistent. It is well calibrated against historical documents and artifacts (from Egyptian tombs, for example) and by tree ring dating. The calibration curve for tree rings extends past 12,000 years in 10 year increments.
using nuclear physics:
Radiometric dating can be performed on samples as small as a billionth of a gram using a mass spectrometer. The mass spectrometer was invented in the 1940s and began to be used in radiometric dating in the 1950s. The mass spectrometer operates by generating a beam of ionized atoms from the sample under test. The ions then travel through a magnetic field, which diverts them into different sampling sensors, known as "Faraday cups", depending on their mass and level of ionization. On impact in the cups, the ions set up a very weak current that can be measured to determine the rate of impacts and the relative concentrations of different atoms in the beams.
The uranium-lead radiometric dating scheme is one of the oldest available, as well as one of the most highly respected. It has been refined to the point that the error in dates of rocks about three billion years old is no more than two million years.
Uranium-lead dating is usually performed on the mineral "zircon" (ZrSiO4), though it can be used on other materials. Zircon incorporates uranium atoms into its crystalline structure as substitutes for zirconium, but strongly rejects lead. It has a very high blocking temperature, is resistant to mechanical weathering and is very chemically inert. Zircon also forms multiple crystal layers during metamorphic events, which each may record an isotopic age of the event. These can be dated by a SHRIMP ion microprobe.
One of its great advantages is that any sample provides two clocks, one based on uranium-235's decay to lead-207 with a half-life of about 700 million years, and one based on uranium-238's decay to lead-206 with a half-life of about 4.5 billion years, providing a built-in crosscheck that allows accurate determination of the age of the sample even if some of the lead has been lost.
Two other radiometric techniques are used for long-term dating. Potassium-argon dating involves electron capture or positron decay of potassium-40 to argon-40. Potassium-40 has a half-life of 1.3 billion years, and so this method is applicable to the oldest rocks. Radioactive potassium-40 is common in micas, feldspars, and hornblendes, though the blocking temperature is fairly low in these materials, about 125°C (mica) to 450°C (hornblende).
Rubidium-strontium dating is based on the beta decay of rubidium-87 to strontium-87, with a half-life of 50 billion years. This scheme is used to date old igneous and metamorphic rocks, and has also been used to date lunar samples. Blocking temperatures are so high that they are not a concern. Rubidium-strontium dating is not as precise as the uranium-lead method, with errors of 30 to 50 million years for a 3-billion-year-old sample.
Use different methods and see if they give the same answer. Cross check those answers against other data, such as the known rate of geological and biological processes such as gene-clocks. Further cross-check assumptions such as the constant rate of atomic decay and constant lightspeed by observing atomic decay rates in distant supernovae. Numerous different methods of calculating this stuff come up with the same answers (within reasonable experimental error) every time.