Does a creature with indestructible still die from having zero or less toughness?

Indestructible only protects a permanent from being destroyed — effects that say "destroy" and lethal damage (CR 702.12b). It does not protect against every way a creature can cease to exist.

State-based actions are checked continuously by the game (CR 704.3). One of those actions (CR 704.5f) states that a creature with toughness 0 or less is put into its owner's graveyard. Critically, this is not destruction — it is a state-based action that simply moves the card to the graveyard, bypassing indestructible entirely.

This is confirmed by CR 704.5f, which makes no exception for indestructible permanents, unlike CR 704.5g (lethal damage), which explicitly does not apply to indestructible creatures.

Concrete example: Your opponent controls a 2/2 creature enchanted with Indestructible. You cast Tragic Slip giving it -3/-3 until end of turn, reducing it to a 2/-1. When state-based actions are checked, it has toughness less than 1, so it is put into the graveyard — indestructible does not save it.

So the key distinction is: indestructible = immune to destruction; toughness ≤ 0 = unconditional state-based removal.