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 Silver is described here. This page will focus more on requirements specific to a living dex (without access to Red, Green, Blue, or Yellow).
You will need access to at minimum three separate save files of Gold, Silver, or Crystal in order to obtain all starters. Since you can breed them, only three files are required.
If you do not have access to Crystal or Mystery Gift, you will instead need four save files, each one of which will need to defeat the Elite Four and meet Bill's grandfather on Route 25 in order to get Pokémon that evolve with the Fire Stone, Leaf Stone, Water Stone, or Thunderstone.
This is just to note that if you have access to a GS Ball Celebi with Crystal, you can obviously trade it to Silver to add it to Silver's living dex. However, if you wanted to, for instance, complete a living dex of 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.