I see a possible engineering problem.
Assuming you wanted to enclose the solar system out to 40 astronomical units, just beyond the semi-major axis of the former planet Pluto, that that would be a sphere with a radius of about 6,000,000,000 kilometers.
The area of the necessary lead sphere would be 4.52 x 10^18 square killometers.
A two meter thick lead shell of this radius would weigh about 2.13 x 10^24 tons.
While substantially less than the sun’s mass of 2 x 10^27 tons it is about equal to the combined mass of Jupiter and Saturn. The problem is, they’re not made out of lead and it could take some time to develop the technology to transmute their current elements into lead. The energy cost for this could also be prohibitive.
There is also the problem in that lead doesn’t have the strength to support a hollow spiracle shape of that radius at that thickness.
You might want to consider something like neutronium. In fact, I’d like to propose an alternative solution. Why not build a ring of neutronium, or some similar material, with walls high enough to contain an atmosphere and rotate it with the sun at the center? At least that way part of the population would always be shielded in the event of a GRB at any given time and you could avoid mass extinctions. Has anyone thought of this?
:o)
“If we can put a man on the moon, we can encase the solar system in lead!”
“Yes we can!” “Yes we can!”
Sounds like you have spent way, way too much time in the bunker with a copy of Excel.
I was just thinking of the Dyson Sphere episode of Star Trek Next Generation where they pull Scotty out of the transporter infinite loop and he rescues them by figuring out the Dyson Sphere around a solar system.