From Bulbapedia, the community-driven Pokémon encyclopedia.
A living Pokédex is a fan term for a game which not only has a complete Pokédex, but has one of every available species of Pokémon stored in its PC boxes as well, usually in their National Pokédex order. It follows the most literal interpretation of the former slogan of the series, "Gotta catch 'em all!". It is considered a holy grail among some collectors. Among others, it is considered a waste of time and box space.
Perhaps one of the most challenging questions is not how to obtain a living Pokédex, since that is mostly a matter of patience and access to multiple games, but rather what to do with the living Pokédex once it is complete. In particular, in this instance, or with any large Pokémon collection in general, the question of storage arises.
In Generation I and Generation II, the Pokémon Stadium games could be used to store large amounts of Pokémon for the purpose of a living Pokédex.
In Generation III, Pokémon Box was introduced, which allowed players to store their large collections without worrying about taking up PC space; in fact, players were rewarded for transferring more Pokémon into Pokémon Box, making the living Pokédex goal more appealing.
In Generation IV, players can use My Pokémon Ranch to store up to 1000 Pokémon, and up to 1500 with the updated version.