Does a player who is eliminated from a multiplayer game take their permanents with them?

When a player is eliminated from a multiplayer game, the game immediately applies a set of rules defined in CR 800.4. All objects owned by that player that are in the game — including permanents, cards in hand, cards in their graveyard, and cards in exile — leave the game entirely.

Ownership is the key concept here. A permanent's owner is the player who started the game with that card in their deck (CR 108.3). So if you own a creature, it leaves the game when you're eliminated, regardless of who controls it at that moment.

If a leaving player controls permanents they don't own — for example, a creature stolen with Control Magic — those permanents don't leave. Instead, they are exiled (CR 800.4a). Any continuous effects that were granting that control end, but since the permanent is exiled rather than returned to its owner, the owner doesn't simply get it back onto the battlefield.

Spells and abilities the eliminated player owns that are on the stack are also removed from the game (CR 800.4). This can have significant ripple effects, as removing spells from the stack may cause other things on the stack to lose targets.

Example: Alice is eliminated while she owns a Grizzly Bears on the battlefield under Bob's control (via Threaten). The Grizzly Bears is exiled because Alice owns it but Bob controls it. Meanwhile, Alice's own Llanowar Elves (which she both owns and controls) simply leaves the game along with her.