A "living dex" is a fan designation for when a player completes the Pokédex and has one of each species in the PC or the party at the same time. It's often considered to be much more difficult and time-consuming than a regular Pokédex because it often involves, for instance, playing the same game multiple times to get multiple copies of a Legendary Pokémon.
The method by which you can get a mostly-completed Pokédex (minus some Gen 1 and Gen 2 Pokémon) on Emerald is described here. This page will focus more on requirements specific to a living dex using only Ruby, Sapphire, and Emerald.
You will need access to at minimum three separate Ruby, Sapphire, or Emerald save files in order to obtain all starters. You will also need access to at least three Emerald save files in particular, and each Emerald save file will need to be able to complete the Hoenn Dex, in order to get all three Johto starters.
This is just to note that if you are attempting to complete a living dex on Ruby, Sapphire, and Emerald simultaneously, you have the ability to breed almost any Pokémon, which can assist in preparing version exclusive and mutually exclusive Pokémon for trades. The Pokémon that cannot be bred are Legendary and Mythical Pokémon and Baby Pokémon.
This is just to note that if you are attempting to complete a living dex on Ruby, Sapphire, and Emerald simultaneously, you will need to prepare to collect Sun and Moon Stones from wild Solrock and Lunatone, respectively, in order to have enough to evolve every Pokémon that requires it. Solrock can be found in Ruby and Emerald, and Lunatone can be found in Sapphire.
This is just to note that if you are attempting to complete a living dex on Ruby, Sapphire, and Emerald simultaneously, you will need at minimum six save files, in order to ensure that each copy of Ruby, Sapphire, and Emerald gets access to each of the Clamperl's evolutions, since only one of the Deepseatooth and Deepseascale can be obtained per game.
This is just to note that if you are attempting to complete a living dex on Ruby, Sapphire, and Emerald simultaneously, you will need at minimum six save files, in order to ensure that each copy of Ruby, Sapphire, and Emerald gets access to each of Latias, Latios, Kyogre, and Groudon. Without access to Southern Island, each game can get only one of Latias and Latios, meaning six files are needed so that all three games can get both.