The technical part is easy, you can get a HB100 doppler radar module under $10. Smaller than a deck of cards, intended for very short-range use but it can measure speed of cars with maybe 100' of range, if you can place it right beside the road. As others have said, my understanding is the bureaucratic and legal parts are next to impossible.
Below is an example of the HB100 signal output after running it through spectrogram software "spek"
Spek – Free Acoustic Spectrum Analyzer / Spectrogram Viewer
The bold vertical lines are a car just passing close by the detector. The thin mostly-horizontal curves show the vehicle speed, where 1 kHz = 32 mph for this sensor. Around 0:45 it also picked up a jogger's legs, which look like around 8 mph. The huge signal at 0:50 is a pickup turning right in front of the detector. The multiple fern-like branches on some close passes, like 0:35 are interesting- I think those are reflections from the individual spokes on the car wheels.