Retro guy says the C64 can't do it.  The lion kingdom says with enough time & the right software, it could.  It would have to use sin/cos transforms instead of a fast fourier transform.  It would take a few disk swaps.  The moon bounce aspect is just a facade for the real ham radio application.  The audio file is 620000 samples, 13 bits per sample. 

https://www.arrl.org/files/file/18JT65.pdf

The lion kingdom's vague understanding is JT65 sends a sine wave on 1 of 65 frequencies to encode 65 possible symbols.  Each symbol is 1/3 of a second.  1 symbol is a sync code.  Every other symbol is a sync code.  The mane problem is detecting the pseudo random sync vector.  They call for a 2048 sample FFT.