What is inside of each one varies greatly depending on the situation. In one chapter it crushes the hospital and contains eight houses and many cupid statues, where Shun and the others have to play shouben kouzou. Later on all the cubes from Japan get together and form a very big one, where the athletic meet-up takes place.
At first only 96 cubes are spotted throughout Japan, but later on they find out there are 108.
Inside each cube there are high schoolers that have to play Kamimaru's games. Here is a list of all the survivers and their cubes.