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.
With Gen 1, the Legendary Pokémon are not a concern, but getting multiple Starters is, due to the lack of breeding.
The method by which you can get a completed Pokédex on Yellow is described here. This page will focus more on requirements specific to a living dex.
This is just to note that you will need to trade for multiple version-exclusives, instead of just one. You will need three Weedle, two Ekans, two Meowth, and two Koffing from your copies of Red and Green/Blue.
Also, if you are playing an International version of Yellow, you will have to trade your only Jynx from either Red or Blue to get Jynx. Therefore, if you are trying to simultaneously complete a living dex in Yellow and another Gen 1 game, be sure to "save" at least one Jynx so that both files can have a Jynx.
You will also need to trade a second Pikachu or Raichu from one of the other games, so that you can permanently keep a Raichu.
The real issue with a Gen 1 only living dex lies in the mutually exclusive Pokémon. To have a living dex, you will need all of the following on your file:
If you have access to Pokémon Stadium, you can get all of these by defeating the Elite Four and Rival.
Otherwise, to complete this living dex on Yellow alone, you will need at minimum four separate save files for Gen 1 in order to get every Pokémon - at least three of which would need to be on Yellow. Your trade process should look something like this: