You can get barium loaded vinyl in sheets which is a sound deadening material. it's used in sound proof rooms. idealy you would make a sandwich with MDF|vinyl|5/8 drywal or mason board|vynyl|MDF
other alternatives could be linolium, modified asphalt roofing materials, sand, lead sheet, liquid filled core. The entire box should not be one box. the layers can not be bonded together. you would in a sense have a box inside a box, inside a box. a lot of unequal and random bracing would help too.

good luck