Find some abandoned or unwanted wood. Make your container. It needs to be "doubled skinned" and the space between the two layers of wood packed full of fibreglass or foam insulation of some kind. Even cheaper use old thick clothes instead.

I helped a buddy of mine make a area to record some songs he had in his extra bedroom/ studio, he had a walk in closet about 6ft x 6ft we stripped it out, and put up foam padding from a foam mattress that we found on the side of the road, it was held up with staples, later on we put up cardboard boxes over the padding held by nails, it was a little weird in there but it worked. When he came up with some cash we stripped all the junk out, put up an extra layer of drywall green board. Stapled more foam on the walls and covered the foam with white fabric. Had a friend who was an electrician install a recording sign with a switch on the inside of the closet and a light on the inside with a switch on the outside. After about a year we got a mic installed inside, instead of hanging it over the door. Hope some of this helps.

You can use nearly anything you want. different surface textures and levels (such as raised areas) will help. This is why you usually see them with walls that looks like they used egg boxes.

good quality Headphones.

Seal the room and then evacuate it - you won't hear a thing in a vacuum. Nice 'n' cheap! :)

There are several ways to limit noise transfer through walls, ceilings and floors. But, since this is a small room, I'll suggest the easiest method I know since it takes up the least space. Furr out the walls and ceiling using 1" x 3" wood furring strips, screwed through the surface and into the studs and joists. Nail a material generically know as mass loaded vinyl barrier to the furring strips using roofing nails. Caulk all seams, perimeters and any electrical outlets as well as any wall penetrations like pipes, etc. with an acoustic sealant. Cover with 5/8" drywall and finish. Mass loaded vinyl is commonly available in two weights, 2lb/sq. ft. (about 1/4" thick) and 1lb./sq. ft. which is about 1/8" thick. I would suggest the thinner material for the ceiling since the 1/4" is so heavy, installation can be a problem. The material should not be nailed directly to the wall. There are several reasons for this: 1. The barrier has to vibrate. The way it works is that the sound waves vibrate the material and as the vibrations propagate through the product, they have to overcome the mass and to some extent, the viscous characteristics of the material. In doing so, the sound energy is quickly dissipated. 2. The barrier is so heavy that it can tear under its own weight. The drywall supposrts it. 3. The drywall adds to the fire rating. 4. Vinyl barrier is just plain ugly. There are many other methods but as far as I know, believe it or not, this is the easiest method to use with existing walls. Stay away from foams, fiberglass and similar sound absorption materials. These products are used to modify room acoustics. They are not soundproofing materials. Neither are egg cartons. They are used to hold eggs.