How do you make a sound proof box?

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.

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