Is it possible to convert radio waves to some small DC power?
Yes, it is possible.  In fact, crystal radios work on this basis.  With a reasonable antenna and a powerful enough AM station, they can drive a high impedance headphone to well-audible volume using only the power received by the antenna.

The basic trick is to us a transformer to step up the low RF voltage picked up by the antenna.  Antennas tend to be rather low impedance, like 75-300 &Omega;.  The output of the transformer would be at a higher voltage, although also at a higher impedance.  The higher voltage is important so that the forward drop of a diode is small compared to the voltage, and therefore doesn't loose too much power when rectified.

In a crystal radio, the secondary of this transformer had a capacitor accross it to form a tuned circuit that resonated at the desired radio station's frequency.  This capacitor was adjustable so that different radio stations could be received.