Not as massive as you think. If you want to fully use the energy output of a star, and avoid that objects in different orbits periodically block each other's sunshine, you need to surrouns with statites. A statite is a structure with a reflective surface that has a sufficiently small mass/surface ratio that the star's light supports the structure against the star's gravity. Such structures always stay in the same place, and don't circle the star in orbits. A statite can still move in any direction around the star, by bending portions of its surface (which means that radiation will exert different pressure on different sections), but once the radiation and gravity force are in equilibrium, it stays put. The star's radiation in this case acts as the cosmic equivalent of bouyancy on Earth. The statite itself must be very thin, but if its made out of carbon nanotubes (remember we are talking about a civilisation that has completely mastered nanotechnology) it can have a reflective and/or photovoltaic layer on the side turned towards the star, and support a large conventional structure, such as a habitat, on the side that's turned away from the Sun. The structure floats on the statite.
How does it work? Photons moving at the speed light have momentum, proportional to their energy.
p = E / c p - momentum E - energy c - speed of light
If light has momentum, this means that it is also able to exert force on an object.
Since F = p / t and P = E / t (t - time F - force P - power), we now have
F x t = P x t / c
F = P / c
The total power of the sun (Psun) is
P_sun = 3,846 x 10 ^ 26 watts (W) (by comparison, the total power of our own civilisation is 1,6 x 10 ^ 13 W, and of all life on Earth 10 ^ 14; only a trillion times less!)
which means it exerts a total radiative force of
F_r_sun = P_sun / c = 1,282 x 10 ^ 18 newtons (N)
In order for the statite to float, the gravitational force and radiative force must be the same
F_g_sun = F_r_sun = 1,282 x 10 ^ 18 N
If we want to find out the total mass the star can support, we put F_g_sun into Newton's universal law of gravitation
F_g_sun = G x (m_sun x m_statites) / r^2 m - mass (m_sun = 1,99 x 10 ^ 30 kg) G - gravitational constant (G = 6,673 x 10 ^ (-11) m^3xkg^(-1)xs^(-2))
Let's assume we want to put our statites at a distance of 50 million kilometers (r = 5 x 10 ^ 10) from the Sun, which is near Mercury. We want to put the statites as close to the Sun as possible, because we want to save on material, but not too close, because in time their surface will be vaporised away.
Using the abovementioned formula we get
m_statites = 2,414 x 10 ^ 19 kg
This is approximately the mass of the asteroid Juno. This is 0,7 % of the mass of the asteroid belt or 0,007% of the mass of Mercury. So the amount of material in Mercury alone alone would suffice to completely surround more than 10,000 Sun-like stars with statites. And nowadays, thanks to Kepler and other exoplanet discovery missions, we are very certain that almost every star (except the old geezers that were formed immediately after the Big Bang) out there has planets and asteroids of its own.
How actually thin the statites need to be? At a distance of 50 million kilometers, the statites cover a surface of
S_statites= 4 x r ^ 2 x pi = 3,14 x 10 ^ 22 m^2
Which gives a mass/surface ratio of
m/S = 7,7 ^ 10 ^ (-4) kg/m^2 = 0,77 g/m^2
If the statite were made from aluminium, that would mean a layer of slightly less than 3 micrometers. Seems quite thin. But we only need tens of nanometers to have a reflective layer or built-in solar cells, which means the rest can be dedicated to carbon nanotubes and the structures they support.
As for mining the planets, Mercury for example, you don't have to dig deep. You can peel like an apple, remove 10 by 10 meters, and then you wait till the new surface cools down. In less than 10 million years (I'm basing this on lord Kelvin's calculation of how much time would the Earth interior require to cool to room temperature; he arrived at a figure of 90 million years; the Earth would have indeed be completely geologically dead only 90 million years after its creation, if it weren't for the uranium core in the center of planet whose radioactive decay has ensured the Earth's core remaining scorching hot for 4,5 billion years; in Kelvin's time radioactivity wasn't known) you should be able to mine the whole planet this way. You only need to find something to do with that material.
