Bob, Timmy, and Susie are playing a Free For All in MTG and Timmy kills Bob by putting him at 0 life. On Susie's next turn, could she play something like Alabaster Potion to make Bob's life total go back up, allowing him to continue playing? Or is Bob completely out of the game? Similarly, when Bob is at 0 life, can Susie still make him take X amount of damage to Gain X life?
A player at 0 life loses the game permanently. Everything they own (including e.g. permanents that are under someone else's control) vanishes from the game; neither the player nor their stuff is a legal target anymore.
From the Comprehensive Rules:
104.3b If a player’s life total is 0 or less, he or she loses the game the next time a player would receive priority.
104.5. If a player loses the game, he or she leaves the game.
If you want more specifics on what happens when a player leaves the game, see rule 800.4.
There are a lot of casual formats to the game, so it would be entirely possible for you to create one that allows players to come back.
However, in all of the official formats (and all of the unofficial one's I've seen) any player who 'loses' the game takes no further part in it, and being reduced to zero or less life causes a player to 'lose' the game.