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. Unlike Pokémon Stadium games and Pokémon Box, players could not use My Pokémon Ranch to transfer large amounts of Pokémon from one game to another very quickly.
In Generation V, there was no storage "game" available. However, trading between boxes could facilitate more trades per session. Also in Pokémon Black and White Versions 2, players could obtain additional rewards for completing the Pokédex.
In Generation VI, Pokemon Bank can store up to 3000 Pokémon online. The ability to transfer many Pokémon between games was restored. Unlike in previous Generations, Pokémon Bank when combined with Poké Transporter is essential to bring Pokémon from the previous generations.