"The low activity of the carbon-14 limits age determinations to the order of 50,000 years by counting techniques."
After 300,000 years, about 1.73 * 10^-16 of the original carbon will remain, which is obviously too little.
To fill in the detail: Carbon-14 has an abundance of about 1 part in 10^12, and one gram of carbon contain about 10^23 carbon atoms, so about 10^11 C-14 atoms. Every single C-14 atom will have decayed in about 200k years.
The uranium-thorium method, the potassium-argon method, and the rubidium-strontium method. Samples containing radioactive beryllium could be dated as far back as 10–30 million years. A newer method of radioactive tracing involves the use of a new clock, based on the radioactive decay of 235 uranium to 231 protactinium.
We'd really like to know why the known accurate methods didn't work in this case, and why they had to fall back to indirect measurements of the flint stones.