What happens in the event of a standoff in Magic?
There are several variants of this situation that are quite different in the rules reasoning and outcome.  In all cases, the situation is a loop.

720.1b Occasionally the game gets into a state in which a set of actions could be repeated indefinitely (thus creating a &quot;loop&quot;). In that case, the shortcut rules can be used to determine how many times those actions are repeated without having to actually perform them, and how the loop is broken.

Case 1: players cannot win or lose the game because of mandatory abilities
The game is a draw.
Example 1: both players have empty libraries and control Dragon Appeasement on an otherwise empty board.
Example 2: one player controls Platinum Angel and Abyssal Persecutor and neither player has a way to remove either from the battlefield.
In this case, the game is a draw.  The game will eventually reduce to each player being able to do nothing but pass the turn.  This is a loop of mandatory actions.

720.4. If a loop contains only mandatory actions, the game is a draw.

Case 2: players cannot win or lose the game because of optional abilities
The game is a draw.
Example 1: both players have empty libraries and control some lands and Words of Worship on an otherwise empty board.
Example 2: both players have some lands and a copy of Beacon of Immortality, but no other way to win.
In this case, the game will eventually reduce to a loop of both players activating the ability that keeps them alive and then passing the turn.  This is a loop of optional actions that spans multiple turns.  The comprehensive rules do not cover the subject, so we need to go to the tournament rules*.  The tournament rules allow for multi-turn optional loops to be continued indefinitely and this causes the game to be a draw.  From the Tournament Rules section 4.4:

Loops may span multiple turns if a game state is not meaningfully changing. Note that drawing cards other than the ones being used to sustain the loop is a meaningful change. If two or more players are involved in maintaining a loop across turns, each player chooses a number of iterations to perform, or announces their intent to continue indefinitely. If all players choose to continue indefinitely, the game is a draw.

Case 3: only one player has an optional ability and is using it to forestall inevitably losing
The player with the optional ability loses.
Example 1: one player has Dragon Appeasement and the other has Words of Worship
Example 2: only one player has an empty library and that player is using Nexus of Fate to prevent themselves from being milled.
This is different from case 2 because only one player is involved in maintaining the loop.  From the Tournament Rules section 4.4:

If one player is involved in maintaining the loop, they choose a number of iterations. The other players, in turn order, agree to that number or announce a lower number after which they intend to intervene. The game advances through the lowest number of iterations chosen and the player who chose that number receives priority.
...
A player may not 'opt-out' of shortcutting a loop, nor may they make irrelevant changes between iterations in an attempt to make it appear as though there is no loop. Once a loop has been shortcut, it may not be restarted until the game has changed in a relevant way. Proposing loops as an effort to use up time on the clock is Stalling.


* The Tournament Rules specify: &quot;Information in this document may contradict (or have information not contained in) the Comprehensive Rules. In such cases, this document takes precedence.&quot;