I want to start a small 250w stealth grow cabinet. This set up needs to be ultra silent and my 250W HPS light hums when it is on. Originally the ballast was in the light itself but I want to make it a remote ballast. With that said I want to build a sound proof/dampening box to place my noisy ballast. the ballast has to go next to my stealth cabinet and will be in my closet. What would be the best way to do make this box? Here is what I had in mind. I want to make a box who's sides are made of drywall I will then use a product called green glue ( its a rubbery caulk like substance that is placed in between pieces of drywall in order to significantly dampen the sound) in between a second layer of drywall to insulate the box and dampen the sound. Any other ideas on how I can silence my ballast?

Drywall is an inappropriate material for grow areas, as it is susceptible to moisture damage. Look at Home Depot for metal grate material, and be certain that all yoru electrical connections are very well insulated.
Don't forget that your ballast needs to breathe. This sounds stupid, but it's true- there has to be a way to dissipate heat, or it will fail prematurely, and may become hazardous.
Silence it by placing it on silicone vibration-dampening feet or a sheet of gel designed to be placed under stereo or other vibration sensitive equipment. I know they sell it though edmund scientific online, and probably at higher end stereo component stores.

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I don't claim to know the first thing about growing anything; hell, I can't make a Chia pet sprout.

by metal grate do you mean mylar over the drywall?

can you give an example of insulating electric connections?

thanks