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 Red 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 two Sandshrew, two Vulpix, two Meowth, and three Bellsprout from your copy of Green/Blue.
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 Green/Blue alone, you will need at minimum nine separate save files for Gen 1 in order to get every Pokémon.
If you have access to Yellow, this number changes to four separate save files - one Red file and three Yellow files. With Yellow, you can collect each starter on each of the three files and trade it to Green/Blue, as well as three of the Eeveelutions and three Fossils, with the Green/Blue file providing the fourth Eeveelution and Fossil. In this scenario, each of the starters traded from Yellow can be evolved to a different form.
Without access to Yellow, your save files would have to look something like the following: