Poke Mart is two words
In other words, we'd need a quadrillion times the space of a terabyte drive to store the atomic information of a Pokémon and many times more that for quantum level storage. And that's not even considering the space needed for any backups that might be created in case of device failure. Or the fact that some Pokémon are many times larger than your average human.
It's at this point that we realize that the Pokémon world must be decades, if not centuries, ahead of our world in terms of computer technology and miniaturization. Especially if you consider the fact that any 10-year-old can buy a Poké Ball from the local
Pokemart for essentially what amounts to pocket change.
==Fail to safe==