The Kitten Box was the home of Oliver and his siblings until, one by one, they were adopted by their future cat owners except him, who was later forced to live on the streets after the box was torn to pieces and dragged into the sewers after it filled up with water from a drain pipe during a thunderstorm; nearly dragging him into there with it until he was saved. The next day, he found a home at Fagin's barge and finally, a home at Jenny's house the very next day.
The box is located somewhere in Times Square in New York. Inside it, there are newspapers for the kittens to walk on and play with.
The box itself is mostly light brown in color with a sign attached next to it; saying: "Kitties Need Home 5.00", later "3.00", and then since Oliver was the only kitten left, he is for "free".