Wow what a fascinating idea! Just thinking out loud here, but the compressive force is proprtional to area, while the boyancy is proportional to volume. So I wonder if making it big enough would allow using some fairly standard materials. What about a cube made of 6 big (huge) sheets of some strong yet light material supported by a sort of skeleton inside?
Or maybe we can use electro-static repulsion to keep it from imploding? Like how a normal balloon compresses the air inside because the rubber tries to shrink but if you had a balloon that tried to expand instead, because of say electric repulsion, or maybe something else, then it would put the air inside at lower than atmospheric pressure.