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 Alpha Sapphire is described here. This page will focus more on requirements specific to a living dex using only Omega Ruby and Alpha Sapphire.
You will need access to at minimum three separate Omega Ruby and Alpha Sapphire save files in order to obtain all starters and both Fossils. Each file will need to progess as far as defeating the Elite Four Round 2 in order to access the Sinnoh starters. Since you can breed them, only three files are required.
This is just to note that if you are attempting to complete a living dex on both Omega Ruby and Alpha Sapphire simultaneously, you will need at minimum four save files, in order to ensure that each copy of Omega Ruby and Alpha Sapphire gets access to Latias, Latios, Kyogre, Groudon, Lugia, Ho-Oh, Dialga, Palkia, Tornadus, Thundurus, Reshiram, and Zekrom.