Does a 0/0 creature with a +1/+1 counter die immediately from state-based actions?

State-based actions check whether a creature has toughness 0 or less, or has damage marked on it greater than or equal to its toughness. If either condition is true, the creature is put into the graveyard (CR 704.5f, 704.5g).

A +1/+1 counter adds 1 to both the power and toughness of the creature it's on (CR 122.1b, 613.4c). So a creature that is 0/0 by its base characteristics becomes 1/1 after the counter's effect is applied. Its toughness is 1, not 0, so the state-based action that destroys creatures with toughness 0 or less does not apply.

Characteristic-defining abilities and counters are applied as part of layer calculations before state-based actions are checked. This means the game always sees the final toughness — in this case, 1 — when deciding whether the creature should die.

Concrete example: You cast Hangarback Walker for X=0, making it a 0/0 artifact creature. Before state-based actions can destroy it, you use Hardened Scales or a similar effect to place a +1/+1 counter on it. Hangarback Walker is now a 1/1 and survives. If you then removed that counter, it would immediately become 0/0 and state-based actions would put it into the graveyard.

In short: it depends on whether the counter is present when state-based actions are checked. With even a single +1/+1 counter, the creature is 1/1 and lives. Without any counters, a 0/0 creature dies immediately (CR 704.5f).