If a creature has indestructible and is dealt damage exceeding its toughness, does it survive?

Indestructible is a static ability defined in CR 702.12b: a permanent with indestructible cannot be destroyed. CR 702.12b states that rules and effects that say 'destroy' have no effect on such permanents, and they cannot be destroyed by lethal damage either.

Damage is still marked on an indestructible creature normally (CR 120.1), and state-based actions still check whether lethal damage has been dealt (CR 704.5g). However, because the creature has indestructible, the game simply does not destroy it when that state-based action would otherwise apply (CR 704.5g references indestructible as an exception).

It is important to note that indestructible does not protect against everything. Effects that reduce toughness to 0 or less, exile, or say 'sacrifice' will still remove the creature, because those do not use the word 'destroy' (CR 702.12b).

Concrete example: Your opponent attacks with a 10/10 Darksteel Colossus (indestructible). You block with a 5/5. The Colossus takes 5 damage, which is marked on it, but at the end of the combat damage step, state-based actions see lethal damage yet cannot destroy it due to indestructible — the Colossus survives untouched in terms of destruction.

Remember: if someone casts a spell that says 'exile target creature' or '-10/-10' to bring its toughness to 0, indestructible offers no protection against those specific effects.