- The usable mass of the solar system is 2x1030 kg (as mentioned by @Alexander) and is completely converted the materials used to construct the sphere. This includes the sun which forms 99.86% of the known mass of the solar system.
- We are building a Death Star-like structure, consisting of internal rooms, rather than a Dyson Sphere or a Hollow Earth shell.
- The structure is made of rooms with average dimensions 10m x 10m x 5m including the wall/ceiling/floor thicknesses
- The walls/ceiling/floor average 1m in thickness and are made of some hollow graphene structure (honeycomb/tuned composite nanostructures/etc.) with an average density of 1720 kg/m3 (the density of the C60 fullerene)
- concrete has a density of 2400 kg/m3; using this will make the structure smaller
- steel anything between 7750 kg/m3 and 8050 kg/m3; using this will make the structure a lot smaller
- All spaces within the structure are filled with air (density 1.225 kg/m3 at sea level).
- We exclude any consideration of water, food, energy generation, etc. (that would depend on the population)
- There is some kind of way that the structure can support itself from the immense internal stresses, e.g. via some kind of force field, and that the air pressure can somehow be controlled (e.g. via airlocks) to prevent the air pressure from getting too high lower down.
- The centre of the sphere is a solid sphere, 1000m in radius, of the same graphene material used in constructing the walls. (I chose 1km, to avoid the walls dramatically diverging upwards, but assuming a hollow 1km core doesn't change the result significantly.)
Given this, we can model the sphere as a series of shells, each 5m thick with a 1m thick "skin" inside that as well as various walls, etc. The total mass of the structural materials making up the shell and core up to shell N is roughly (1.684x105N3 + 2.026x108N2 + 4.1x1010N + 7.2x1012) kg. The mass of air contained within all shells up to shell n is roughly ((16.8N3 + 7.66x104N2 + 9.03x106N) kg.
This gives us a total of (1.6846x105N3 + 2.03x108N2 + 4.1x1010N + 7.2x1012) kg.
For a total mass of 2x1030 kg, and solving for N, we get about 228,127,000 shells for a total radius of 1000+228127000*5 = 1,140,636,000m = 1,140,636km. This is just under twice the radius of the Sun (695,700km), and nowhere near the orbit of Mercury, nevermind the asteroid belt.
If you assume that the entire structure contained no air, it would make little difference to the calculation as the air is about 0.5% of the mass.
Isaac Arthur has done a YouTube video on a similar concept, focusing on shells of earth-like surfaces (starting at about 8:04 in, though you might want to watch it from the beginning for some context).