Well, that depends on what kind of structure do you need and how you build it...

Let's say it's a ring made of nanites (which sort of solves all material tension problems, because we assume that nanites can automatically rebuild any damages). Let's assume that the radius of the ring is 50000 light years (an approximation for the Milky Way radius, which equals 5*10^19 m), its width is 1 km (10^3 m) and its thickness is 100 nm (a size of a nanite, 10^-7 m). Let's assume that the nanites are made of carbon and have its density, which equals 3*10^3 kg/m^3.

The mass of this structure is going to be:

M = ro * V = ro * S * L = ro * h * w * 2* pi * R = 3*10^3 * 10^3 * 10^-7 * 2 * 3 * 5 * 10^19 = 9 * 10^19 kg.

Milky Way mass is 10^12 solar masses, and the mass of the Sun is 10^30 kg. In fact, the mass of the Moon is 10^22 kg. So one Moon is more than enough to build such a ring around the Galaxy...