Construct the box with materials that are good sound-absorbers. A few of those are:
-- styrofoam -- thick cloth -- air (trapped between two layers of foam or glass) -- composite material, made of layers of foam and mylar -- "mass loaded" vinyl
There are also geometric methods of baffling and insulating, that work by building specially shaped grooves and nooks and crannies into the surfaces. But you're probably not equipped to get carried away like that for a box. Your best bet is simply to construct it out of thick sheets of foam. You could use the fillers that come in the cartons used to pack computers, appliances, DVD players, etc. Or you could use several layers of egg-carton.
To make a soundproof box, use materials like styrofoam, thick cloth, or mass loaded vinyl to absorb sound. Construct a sturdy box and line the interior with these materials, sealing edges to prevent sound leaks. Testing the box will help you determine its effectiveness in soundproofing.
;