I see your line of thinking. Many foams (especially 'open cell' type) have acoustical aborptive properties. Unfortunately, mechanism of acoustic transmission through a partition is a different mechanism. The overwhelmingly dominating parameter affecting the ability of a partition to block sound (at lower frequencies at least) is its mass. A heavy brick wall is a better blocker of sound than a sheet of paper.
At higher frequencies, the stiffness of the material may start to have an effect, but this is secondary.
The absorptive properties of the material have very little to do with it (except in the sense that lining the interior of the enclosure with foam would reduce the overall sound level inside the box, thus reducing the level of sound energy which could be transmitted to the outside).
Another critical point is that the enclosure must be air tight. A double glazed window is good blocker of sound, but only when it is closed!
Here's an idea for a small 'sound proof' enclosure. Line an small domestic refridgerator (one of those small beer coolers should do, although maybe not one with a glass window) with open cell foam / glass fibre mats. This should provide a nice air-tight seal, as well as degree of sound proofing. You could then construct a masonry box around it. For access, leave one side open and cover the hole by bolting several (maybe 5?) layers of 15mm gypsum board. Shouldn't cost more than 300 dollars or euros. You may find that the fridge on it's own is enough for you. Or you could place the small fridge inside a larger one (chest feezer?). If all this is too small for you, there are plenty of companies which sell larger acoustic enclosures; The kind of thing you might find on a manufacturing floor with a machine operator inside.
Hope this gives you some ideas.
M
