A rigid shell of the Dyson sphere might have a thickness of a few feet, depending on the strength of the material fabricated. It would also have to rotate to make artificial gravity. To maintain habitable temperatures the sphere would need to be bigger than Earth's orbit.
To avoid the dynamical stress a solid shell might undergo, a Dyson Sphere might be a constellation of many small independently orbiting structures - like squares of mirrored glass on a disco ball. The energy-collecting elements would likely be very thin, while habitat segments would be thicker. Their orbital paths would be adjusted by using solar sails or ion engines.
The classic science fiction story "Ringworld" By Larry Niven, meticulously describes a spinning Hula-Hoop type structure, rather than a sphere as imagined by Dyson.
At least a partial Dyson Sphere - or Niven's ring - around the sun could be built from dismantling the planet Mercury and reassembling it into shell segments. The problem is that the energy required for destroying a planet is 100 billion times the U.S. annual energy consumption (sheesh! Even the Death Star probably didn't carry those kinds of batteries!).
So where would that energy come from? The sphere would have to be built piecemeal with the energy collected from the first segments being use to fuel further planetary disassembly.
An army of robots would have to do the task. They would need to use resources to build more robots - like the enchanted brooms in Walt Disney animation of Paul Dukas' symphonic poem "The Sorcerer's Apprentice." Even with this bootstrapping approach, the construction would take centuries because orbiting solar collectors can only capture so much energy over time.
There have already been astronomical database searches for Dyson spheres. The spheres would absorb and re-radiate the star's energy as infrared light. As seen from Earth, a shell or partial shell would glow at a comparatively cool few hundred degrees Fahrenheit.
Galactic archeology is now being done by scouring infrared all-sky databases for sources in this temperature range. An artificial structure would need some other clues, perhaps an unusual spectral signature not found in a dust-shrouded young stellar object, or a complex, repeating fluctuation in brightness that is hard to explain by normal circumstellar dynamics.
Finding unequivocal evidence for a Dyson Sphere would tell us that there are no practical limits to the capabilities of an intelligent species, given time, perseverance, and a godlike mastery over matter and energy.
Image credit: NASA