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 on Crystal is described here. This page will focus more on requirements specific to a living dex.
You will need access to at minimum three separate save files of Gold, Silver, or Crystal, across two different cartridges, as well as three save files of Red, Green, Blue or Yellow (two if you have access to Yellow) in order to obtain all starters. Since you can breed them, only three files are required in both cases. Your trade process should look something like this:
If you are playing an original Game Boy Color cartridge copy of Crystal, and if you are able to connect to Pokémon Stadium with your copy of Red, Green, Blue, or Yellow, then you only need a single save file on Red, Green, Blue, or Yellow. This is because you can obtain the starters and Fossils by defeating the Elite Four and Rival.
The 3DS Virtual Console rereleases of Crystal provide access to Celebi with the GS Ball item. This is just to note that Gold and Silver do not get the GS Ball. If you wanted to, for instance, complete a living dex of Gold or Silver and Crystal simultaneously, you would need to get multiple Celebi in Crystal, requiring multiple save files that will each need to defeat the Elite Four.