So let's say you have a Dyson sphere that is 5 kilometers thick . Now, it isn't solid. It's meant to be inhabited. There's the inner shell which is completely covered with solar panels and any other manner of power generation facing the star. It's 500 meters thick. There's the outer shell which again is 500 meters thick, but is covered in armored plating, defensive weaponry, and the occasional hanger door. Between these two shells is an empty 4 kilometer area that is FILLED with breathable air. People living in the sphere would inhabit centrifugally spinning towers that act as support columns between the shells, but I digress. Where would one go about getting all that air? This is assuming that the creators of the sphere are humanesque (more or less), and require the same air we do.
You Must Find a Source Outside Our Solar System
Given the size of a Dyson sphere, it's fair to say that you need a LOT of gas.
An idealized Dyson Sphere is large enough to contain planets such as Earth and, per Wikipedia, would be about 1AU in size. Given that such a sphere CONTAINS our planets and their orbits, it's intuitive that you could not possibly harvest an amount of elemental oxygen necessary to create a breathable atmosphere.
Let's say though that you wanted only to build your Dyson Sphere around the sun, leaving the planets cold (or transmitting directed energy to them from your Dyson Sphere). That's much smaller. Your challenge is that even Jupiter - our largest planet - could fit inside the sun 1,000 times over. In fact, the sun comprises 99.8% of the mass of the solar system itself!
In that scenario, if you wanted to fill it with air from this solar system you are trying to build a sphere around a body with 99.8% of the solar system's mass that is 4km thick and fill it with breathable air from the remaining 0.2% of available mass in the solar system, only about 5% of which is actually oxygen. That 5% number also needs to contribute to at least 19.5% of the atmosphere of your creation.
Long/short - There is not enough elemental oxygen in the solar system to allow a Dyson sphere with a 4km atmosphere of breathable air. You must obtain your oxygen from another solar system; quite likely MANY other solar systems.
The builders would plunder the star's planetary system (and neighbouring systems, as needed) for raw materials when building the Dyson shell.
Without knowing the exact dimensions it's impossible to say how much gas is needed to fill the inhabited gap, but assuming "air" here means a mix of gases – nitrogen, oxygen, carbon dioxide, etc – comparable to Earth's atmosphere, these gases can be harvested from the atmospheres and satellites of gas giants and ice giants and the crusts of terrestrial planets:
Electrolysis, photolysis and other methods will be needed to wrest the desired gases from their bonds (such as nitrogen from ammonia, oxygen from ice water and iron oxide), but if you're building something the size of a planetary system that should be well within your ability.
Bonus: The builders would end up with a lot of free hydrogen and helium, which they could put through an artificial neucleosynthesis (fusion) process, to produce needed elements.
So the good news is that if you have the capability to construct a Dyson sphere, you have the capability of supplying it with resources.
The sheer amount of matter needed is huge, and you'd need to steal the planets and other debris from many other stars to get enough. And since you're stealing all that stuff you can get their gasses too.
If you don't like the idea of moving dozens of solar systems worth of planets across light years of space, the other option is to make an energy to matter converter. Then you just need a whole lot of energy, which can be provided by the sun.
Best course would be to collect the asteroid belt, kupiter belt, and any other stray rocks that are floating around, turning them into a Dyson swarm with as many energy collectors as you can put up, and start feeding it all to an e->m engine. Use that to expand your collectors until they form a Dyson ring, and then enclose the globe.
This would be an exponential process, since the more you enclose the more energy you have to work with.
At this point you've got your inner shell, and you should have all the energy you need to finish the outer shell and then start making atmosphere to fill it.
Change the ring's size and you'd have plenty of oxygen.
One earth volume (1,097,509,500,000 km^3) is plenty large enough to construct a Dyson-esque ring that could support human life if you were to shrink the design to a spinning ring that exists in a Goldilocks orbit the diameter of Jupiter and a width of about 1/3 of the diameter (45,000 km about) tilted at a sufficient angle and spinning to provide centripetal gravity. You wouldn't want the depth of the ring to be too great, as the more material you pack in the greater the force pushing against the outside of the ring causing structural instability. Limiting the depth to 50km or less is sufficient and beneficial, allowing for bodies of water to naturally form while reducing the impact to the tensor stress. That being said, you'd still need an incredibly strong material to keep the ring structurally intact; but carbon microfilaments have come a long way and could be woven into a form strong enough (theoretically).
The biggest challenge would be keeping the air in. Our atmosphere extends up to 10,000 km; however most of what we consider important is limited to the first 33 km. I am not certain it would be sufficient to build a 50 km wall, I'd prefer something closer to 5,000 km with a 30 degree angle (4,000 km width) to be safer.
Still, that being said we're "only" talking a net 315,028,000,000 km^3 of material. Heck, you can almost build 3 of them from 1 earth-sized planet. Granted it wont be the material you necessarily need but I have no doubt gathering enough metal from our nearby asteroid belt would do the trick.
The key point would be to construct a whole lot of pre-formed wedges of the shell and then set them together all at the same time like when you put the keystone up for a bridge or archway. Once the framework is built, that's when you can start adding the silicate and carbon material on the inside while frequently checking for stress points using thermal and laser imaging. If stressors exist, additional support can be added to the frame. Meanwhile, keep in mind that if you build a mountain you'll need to offset that added weight with something on the opposite side of the ring so as to avoid wobbling.
All that being said, the surface area of such a sphere would be approximately 10x of earth's so you would need to basically raid Europa and Titan for their water. Combined, those two bodies contain over 11 times the water Earth does. You could also raid the asteroid belt for another billion liters. All that water would provide you the oxygen you need to support life on the planet; particularly after you add carbon to build upon the greenhouse gases necessary to heat the ring's atmosphere. Excess hydrogen could be used up to fuel the enormous fleet of construction vehicles building the ring in the first place.
Fusion of Hydrogen
All elements below Iron on the periodic table are created in stars by a fusion chain that starts with Hydrogen. If you've got the technology to build a Dyson Sphere, you've got the technology to fuse lightweight elements.
Collect the host star's solar wind - you probably need to do that anyway, if your sphere is solid and completely encloses the star - and fuse it up to the appropriate elements.
The largest source of oxygen in the Solar System is the Sun.
About 1% of the Sun's mass is made up of Oxygen. About another 0.5% of the mass is a mix of Carbon and Nitrogen.
Given the super advanced tech needed to craft such a large structure, harvesting the Sun for material should be well within the civilization's abilities.
If you want to limit the construction material to what is available within the Solar System, then you just calculate how much area your spherical shell will contain for its atmosphere, then you adjust its radius until the area matches up with the available material.
I suspect it will be within the distance of the Earth's orbit, so a combination of shades and radiation fins will be needed so that the megastructure doesn't become an oven.
Starlift a late stage star.
At a radius of 1 AU you need about 2.81e26 cubic meters of solids (mostly Iron, Carbon, and Silicon I would guess) and 1.12e27 cubic meters of air. The Earth itself has a volume of about 1.10e21 cubic meters. This means it would take about 100,000 Earths to build such a mega structure.
That said, there is one source of heavy element (stuff above helium) that is WAY bigger than 100,000 Earths: pre-supernova stars. Just before they go supernova, a large star builds up an iron core that can reach up to 1.4 solar masses, and similarly massive amounts of Oxygen, Nitrogen, etc. This is WAY more than you will find in any single planetary system, and about 5 times as much material as you need. Once you've mined the star of all of its precious heavy elements the star itself will be lighter, cooler, and less explody... a perfect spot to build a brand new Dyson sphere around.