>>58144558
Pokemon don't turn into data, they just shrink.
You may be thinking
>Okay, smartass, but then how do you trade them through link cables, which are stated to be canon in-world objects?
It's simple, really. You attach the link cables to your Gameboy and your friend's, and then the Pokemon shrinks down and crawls through the cable to go into the other Gameboy.
See picrel.
>Okay, but then how do Pokemon get stored in the PC?
They don't get stored IN the PC. They shrink down and crawl through the ethernet cable, and then come out at some other location.
Since it's Bill's PC, I assume Bill keeps everyone's Pokemon in the secret garden behind his house.