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 completed Pokédex on Sapphire is described here. This page will focus more on requirements specific to a living dex.
You will need access to at minimum three separate Ruby, Sapphire, or Emerald save files in order to obtain all starters. If you do not have access to Colosseum or XD, 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.
You will also need access to at minimum three separate FireRed or LeafGreen save files in order to obtain all Kanto starters. If you do not have access to Colosseum, you will also need to progess as far as defeating the Elite Four and delivering the Sapphire to Celio on One Island in each file to unlock the Johto Legendary Beasts.
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, Baby Pokémon, Nidorina, Nidoqueen, Nidorino, Nidoking, and Unown.
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. Additionally, wild Clefairy in FireRed and LeafGreen can hold Moon Stones.
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 Ruby, Sapphire, or Emerald save files, in order to ensure that each copy of Ruby, Sapphire, and Emerald gets access to each of Clamperl's evolutions, since only one of the Deepseatooth and Deepseascale can be obtained per game. If you intend to complete living dexes on all five games, you would need ten save files.
This is just to note that if you are attempting to complete a living dex on all five games simultaneously, you will need at minimum five FireRed or LeafGreen save files, in order to ensure that each copy gets Mewtwo and the Kanto Legendary Birds. Although the birds can be obtained in XD, Mewtwo cannot.
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 nine FireRed or LeafGreen save files, in order to ensure that each copy of Ruby, Sapphire, and Emerald gets access to each of the three Legendary Beasts, since they cannot be bred. If you have access to Colosseum, you can simply use three Colosseum save files instead. If you intend to complete living dexes on all five games, you would need five Colosseum files or fifteen FireRed and LeafGreen files.
This is just to note that if you are attempting to complete a living dex on all five games simultaneously, you will need at minimum ten Ruby, Sapphire, or Emerald save files, in order to ensure that each copy 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.