Short-range radar and LIDAR, how do they work?
I don't think this site is a place for comprehensive technology reviews or dissertations on microwave-based technology. In short, light travels in air at 12 inches per 1 ns. So this is not completely out of touch for modern electronics to detect sub-ns delays. 

Here is one device, after a 30-second search, TI AWR1443 single-chip radar. So the answer is that radar processors avoid ultra-fast sampling rates by "baseband processing" (aka "quadrature demodulation") of the radar carrier signal (70-90 GHz range) into more manageable frequency range, and then uses a 200-MHz ARM processor to get the final result.