What happens when both players run out of cards with Gideon of the Trials and an emblem in play?
There are no specific rules to handle this kind of situation. If both players can't lose the game, play just continues as normal until one player can lose and does, or somebody concedes. The tournament rules do allow for player to agree to declare the game a draw, and doing so may be an attractive option in the situation you describe.

In addition, in a tournament, if a game does not end, eventually the tournament organizers will announce that time for the round has ended, and that games can proceed for a total of five more turns. If the game has still not ended at the end of those turns, it will be declared a draw.

If a player resolves an Anguished Unmaking targeting their opponent's Gideon, then the next time that opponent would lose the game, they actually lose the game.