Yes, this could be done. The structure would be a very thick band (from a construction point of view), or a very narrow disk (from an astronomical point of view).
Mild steel has a yield strength of about 250 MPa, and a density of about 8,000 kg / m³. For a structure close to the earth, the weight is about 80 kN / m³, or 80 kPa / m of the band's vertical thickness.
The band needs to resist a hoop stress of (the band's weight converted to a pressure) times (the radius of the hoop) divided by (the thickness of the band). The previous paragraph shows that mild steel can handle this stress if the radius of the hoop is less than 3,000 times the vertical thickness of the band. Extra band thickness will provide a margin of safety. The earth's diameter is about 12,800 km at the equator, so the band's vertical thickness should be about 2,200 meters times the margin of safety.
The band would have to be fairly wide horizontally, in order to prevent buckling. If the band were just one meter wide, the top would flop down, and it would no longer be tall enough to hold up its own weight. I have not done the calculations, but I expect that if the width were at least 1/6 of the height, vertical buckling would not be a problem. Waves along the length of the band might be entertaining, though.
Cyclic loading and stress-corrosion fatigue would also be issues. I have not done these calculations either.
So a belt of mild steel that was 1 kilometer wide by 6 kilometers tall by 80,450 kilometers around would probably suffice. That is about 500,000 cubic kilometers, or 4 million billion metric tons. Nearly all of that material is iron, which is fortunately abundant on Earth. About 0.3 percent of that material is carbon, or 12,000 billion metric tons.
Mass of earth's atmosphere: about 500 million million square meters times 10,000 kg per square meter, or 5 million billion metric tons. Thus, even extracting all of the carbon from the atmosphere would yield only about 800 billion metric tons of carbon. Extracting all of the world's proven coal reserves would only yield another 800 billion metric tons of carbon.
If we reduce the band size to 3,000 meters tall by 200 meters wide, and use ridges to prevent buckling, we reduce the material requirements to 30,000 cubic kilometers. This consists of 250,000 billion metric tons of iron and 750 billion metric tons of carbon.
Stronger materials would allow using less material, but would make fatigue issues more worrisome. Unfortunately, the alternative materials might be even harder to find than carbon.
As the MITSFS pointed out, "The Ringworld is unstable". This problem can be dealt with by leaving some of the jacks used to raise the ring in place. Unfortunately, we run into the problem of supporting the weight of the jacks. If the jacks are made of mild steel, they must either be only 3,000 meters tall, or must grow exponentially at the base. Alternative materials would be very useful for the jacks.