The punishment box was a means by which Alixus convinced members of her community to punish those who broke agreed-upon rules and guidelines. It was a simple device, cramped and allowing little to no air inside. Those who broke the rules were placed in the box for a period of one day or longer, depending on the crime.
Stephen had been placed in the punishment box prior to the arrival of Benjamin Sisko and Miles O'Brien after he stole a candle from one of the villagers. When he was released, he had to be fed water as he was too weak to drink it himself.
When Sisko resisted Alixus' attempts to assimilate him into her technology-free way of life, a struggle embodied by his refusal to change out of his Starfleet uniform, he was placed in the box. He was eventually released and brought to Alixus, who offered him food and water if he would change into new clothes. Sisko refused and re-entered the box of his own free will. (DS9: "Paradise")