Not impossible, just extremely difficult. I'm a ham and getting some contacts over moonbounce is a personal goal of mine. Historically this kind of thing has required some pretty large antenna arrays and very high power though:
Isn't there a moon bounce mode in WSJT (or one of those digital modes) that provides enough coding gain that 100W and a single large Yagi is enough? I seem to recall hearing something like that... but, yeah, on CW a monster antenna and the legal limit of 1500W seems to be the median system.
A long time ago I started collecting parts for a 432MHz EME system. Life got in the way and I never built it out. Good luck with your endeavor!
> but, yeah, on CW a monster antenna and the legal limit of 1500W seems to be the median system.
A good ten years ago or more, they used Arecibo to transmit CW moonbounce on 70cm. I was able to receive it in my back garden with a handheld and an 11-element Yagi balanced on my clothesline ;-)
The big dish antenna at Stanford University (visible from I-280) was, among many other things, used to monitor Soviet radar signals from Sary Shagan in Kazakhstan, that bounced off the moon some of the time.
which has this tidbit which explains why it works as well as it does when it works:
"Fortunately for us, the moon appears only slightly rough to radio waves; most of the reflected energy comes back from an area at the near point just a few miles in diameter. The bulk of the energy striking farther around on the side is reflected out into space and never returns to earth."
It's been being done for 70 years or so. You just need to be able to generate quite a lot of power on VHF upwards, and have a very low-noise antenna preamp, and a large array of directional aerials to focus the signal.
If you're reasonably handy with simple hand tools you can build a moonbounce array for a couple of thousand and a month or so of evenings.