Hi EF, EME is achievable with home-brew gear, there are quite a few Hams who have EME capable stations.
Probably the easier path is to get your ham ticket which will allow you to transmit a reasonable power level on a frequency where someone else is listening.
Guys like vk3um ve4ma are some of the leaders in the field.
There appears roughly 2 paths, first go for morse or voice level communication which needs a lot better S/N ie more power, better antenna (dish) and better receive path. The other path is to use spread spectrum techniques (computer encoding decoding) using a program eg jt65, S/N can be as low as - 18dB ie below noise and still get decode. This allows medium level eg 100w, modest antennas ie yagis, and little preamp. Activity on 140MHz JT65 ham band is quite active, building a receive only station negates licencing issues. Another area that may interest you is WSPR, another digital mode. Communicating across the globe with 0.1 W is fascinating.
It is an interesting technical 'hobby'.
Regards Rob