[quote=dot80]
We couldn't keep and transfer an unlimited number of Pokemon through a computer. We couldn't hold a gargantuan Charzard in a minuscule pokeball (we really couldn't hold any pokemon in a pokeball).
[/quote]

I'm not sure how canon this is, but apparently Pokeballs work by "downloading" the target Pokemon, like a USB. Pokemon are stored as data rather than as living creatures and the information is loaded when the Pokeball opens.
The Pokeball is the same thing as a pen drive - except it stores one creature rather than pictures and documents.

And this tech is in the making today. It is now possible to take memories and store them in a kind of DNA USB. It is possible to take ones memories, store it as data and then convert it into a usable form, albeit it is insanely expensive. It's about 86 million US dollars to turn 6.8 GB of DNA into binary and then convert it into a usable form.