It would be impossible for us to build a solid sphere around a star. Freeman Dyson admitted as much. He proposed instead a concept in which we released many independent machines to circle the sun, collecting energy and beaming it to Earth.
The satellites might be arranged in neat rings. Or they may zoom at different orbits around like a swarm of bees gathering the pollen of power. Some might be habitable, or they might be used solely for energy gathering.
Instead of a solid sphere or ring of satellites, there's a third configuration that might make up a Dyson sphere. In this instance, a number of solar sails would orbit the sun. These sails would create a loose bubble around the sun, held in place thanks to a balance of gravity and radiation pressure from the star within.
These satellites would actually be called statites (derived from the words satellite and static) because they'd hover in one place instead of moving in orbit. As with the ring or swarm concept, statites would absorb outgoing solar power and then redirect it to Earth for our use.
Regardless of the final blueprint, a Dyson ring, sphere, bubble or swarm is going to require material resources and energy on an unprecedented scale. We won't find enough raw materials for this project on Earth. So some speculative types have proposed an alternative — harvest the materials from other planets, such as Mercury.
Like an inconvenient wetland bogging down a tract of land zoned for commercial use, we could simply disassemble Mercury and put its sun-scorched surface towards a better purpose. That's an idea proposed by Oxford University physicist Stuart Armstrong.
Mercury is loaded with useful materials (such as iron) and it's the closest planet to the sun, so it makes sense to start there. Once Mercury is dismantled and the first Dyson rings are installed, the project would gather momentum and speed, collecting more and more energy, fueling the production of larger and better solar collectors. And of course, all of that energy would be put to use for other purposes, such as supercomputing (on a never-before seen scale), faster space travel technologies and innumerable other ideas we haven't even conceived.
We wouldn't need to do all of this with painstaking labor. Robots would perform the work of mining and assembling new solar arrays. And the number of robots would continually increase (thanks to self-replicating capabilities) as the scale of the project grew, until they became an army of automatons crushing planets and asteroids to manufacture evermore energy collectors.