Start out by doing some basic calculations. Air is mostly N2, with a atomic mass of 28, and O2, with a atomic mass of 32. There is more nitrogen than oxygen, so let's say air has a average atomic mass of 29.
At 0 °C and 1 atm, one mole of ideal gas occupies 22.4 liters. Let's say 20 °C is more realistic, so we'll scale that by the absolute temperature ratio (22.4 l)(293 °K)/(273 °K) = 24.0 l. So the density of what we will call "air" is (29 g)/(24 l) = 1.2 kg/m3.
So for every cubic meter you can enclose, you get to use 1.2 kg of material. That by itself is not a big deal. The gotcha is that the surface has to somehow be able to withstand about 15 pounds per square inch, or about 11.4 tons per square meter, or 101 kN/m2.
Let's say you want to make a sphere 10 m in diameter. That has a volume of about 524 m3, so you can use up to about 630 kg of material.
It should be pretty straight forward to calculate the thickness of various materials required to make a 10 m diameter sphere than can withstand 1 atm of external pressure. None of the materials we have today come close.
However, that's a really naïve way to make a vacuum tank. A suitable internal lattice structure will save well more than its own weight in weight of the skin material. The real engineering issues are with the clever design of the internal supports. The optimum arrangement of internal supports depends on the relative strengths and densities of the available materials, so you can't just create one design then work backwards to see what the material properties need to be.
To really answer the question, you'd have to pick some material properties, iterate to find the optimum structure design, see where you're at, adjust the material properties, adjust the structure to the new tradeoffs, etc.
Note also that scale matters. A sphere with twice the diameter gives you eight times the mass budget, but only four times the surface area. There is obviously some win in going larger. On the other hand, larger diameters give you less advantage from the arch effect due to the curvature, and the interior supporting structures don't scale the same way.
In the end, I expect that this is solved with a multi-dimensional iterative design process.