In this Astrophysicists professional opinion it is unlikely (and probably impossible) to construct a rigid "shell" structure which is able to hold itself over the sun, or even hold itself apart from it's own gravity field before collapsing into rubble (as another poster stated). If you did construct such a structure it would also be unstable and prone to falling into the sun, ala Ringworld, but if you could construct such a structure in the first place that may not be an issue.
The simple fact is that the stiffness/density ratio to withstand the gravity of a sun is enormous, probably impossibly so. Also, that sort of structure would probably have an enormous mass.
A very popular way of solving this is the "Dyson Swarm", which other posters have mentioned. Just keep building and launching normal satellites until they literally block the sun. An alternative (one which is not mutually exclusive to the swarm approach) is to build a structure that does not need to withstand gravity. Instead of a shell build a thin membrane that surrounds the sun, light enough that the solar radiation pressure that object feels is slightly larger than the gravitational pressure. Instead of tending to fall into the sun the entire balloon would inflate out from the sun until it was taut. The structure would then only have to withstand the tensile force of the excess solar radiation pressure, so lets say 1% of the gravitational force, and tensile force can be withstood with lighter materials than compressive force to boot.
So what would the areal density of such a membrane have to be?:
Pressure * MembraneArea = MembraneMass * Gravity
MembraneMass/MembraneArea = ArealDensity = Pressure / Gravity
Gravity = SunMass * GravitationalConstant / radius^2
Pressure = FluxDensity / c (assuming that our membrane is perfectly absorptive, also note that we don't need to take into account the pressure of the photons leaving the membrane, as they will be split evenly between the inner and outer surfaces and cancel each other out.)
FluxDensity = SolarLuminosity / MembraneArea (//*The SolarLuminosity is the total power output of the sun.)
MembraneArea = 4 * pi * radius^2
so: ArealDensity = (TotalPowerOutputOfSun / (4 * pi * radius^2) / c) / (SunMass * GravitationalConstant / radius^2)
The radius cancels out! That means that the same membrane (barring heat constraints) can be used anywhere in the solar system!
ArealDensity = (TotalPowerOutputOfSun) / (4 * pi * SunMass * GravitationalConstant * c)
ArealDensity = (3.839E26 Watts (kg*m^2/s^3)) / (4 * pi * 1.9891E30 kg * 6.673E-11 m^3/kg/s^2 * 3E8 m/s)
ArealDensity = 7.67E-4 kg/m^2
So all we need to do is make a very thin structural membrane, line it with incredibly efficient solar cells, as well as efficient transmission to the laser stations studded every few tens of thousands of square kilometers, into a sheet of membrane that masses around 7 grams a square meter! (safety factor, as well as extra to hold up those laser installations) Easy peasy, that's just an order of magnitude less than a sheet of ordinary paper! For an even more relevant example this paper references a current deployed solar array areal density of 80 g per square meter. Coincedentally enough that's actually exactly the areal density of a sheet of paper, so an order of magnitude of improvement is actually what we are trying to achieve.
As far as the total mass of this system, that's ArealDensity * 4 * pi * radius^2. Let's think really grand and build it 10% past Saturn. .0007 * 4 * pi * (1.1 * 1.43E9)^2 = 2.18E16 kg. That's only 3.6E-9 the mass of our planet, or 2.3E-5 the mass of Ceres, so once we get Asteroid mining started up that'll be no problem. Heck, if you wanted to be lame and build it at 1.1 times Earth's maximum distance from the sun you could make it more than 100 times lighter than the Saturn variant.