With the objectives: (1) to increase the surface area available for population, and (2) to collect all the radiant solar energy; a spherical surface is constructed around the Sun from the material of the solar system.
The primary objective requires the surface to be habitable. The habitable zone of the Sun is at radius from 75,000 Mm (½ AU) to 450,000 Mm (3 AU) – a radius of 150,000 Mm (1 AU) would be ideal. These dimensions give a surface area of in the range 70 billion Mm2 (140 million Earths) to 2500 billion Mm2 (5000 million Earths) – ideally 280 billion Mm2 (550 million Earths). The thickness of the sphere depends on the radius chosen and the volume of material available.
The Sun has a mass of nearly 2000 billion trillion Gg (333,000 Earths). The rest of the solar system is about 1.4‰ of the Sun’s mass, giving 2.75 billion trillion Gg (466 Earths) of material for construction. Saturn is the least dense planetary object in the solar system with a density of 0.0007 Gg/m3. This density of material would give a range of thicknesses from 56.8 metres at ½ AU to 1.6 metres at 3 AU – 14.2 metres at 1 AU. This density of Earth is 0.0055 Gg/m3 which would give a range of thicknesses from 7.2 metres at ½ AU to 0.2 metres at 3 AU – 1.8 metres at 1 AU.
This may not sound like the snappiest line from 2001: A Space Odyssey (1968), but it evidently caught the imagination of John Landis, who has worked references to a mythical film of this name into most of his own movies - memorably as the grotty British skinflick watched by an assortment of lycanthropes and zombies in the climax of An American Werewolf in Paris [sic] (1981).
Ghastly Beyond Belief, Neil Gaiman and Kim Newman