An O'Neill Cylinder is basically a thin walled tube with end caps. Bursting pressure for the cylinder portion is: yield strength of the material x material thickness divided by the radius of the tube. Spheres can take twice the pressure if everything else is the same, or use half the thickness at the same pressure.
So, assuming standard earth sea level air pressure (14.7 PSI), and an equal amount of internal mass mounted on the cylinder, plus a little for uneven pressure and safety factors, the pressure on the shell would be about 40 PSI. Using lunar aluminum at 40,000 PSI yield strength, a 40000 in radius (slightly larger than 1000 m or 1RPM spin rate for 1 g) cylinder would need to have a skin of 40 inches. Note that the shell pressure includes the weight/unit area of the skin material.
There is another limitation on the size of the cylinder though. The circumference of the cylinder cannot be longer than the breaking length of the material (en.wikipedia.org/wiki/Specific_strength). So the cylinder made out of this medium grade aluminum could be no more than ~10km in circumference (or 3100m in diameter). As an engineer, I'd want some margin on that, so I'd keep the 1000-1200 m radius as my limit - not a particularly large OC.
Also, due to spin instability, single OC's can't be ridiculously long compared to their length. If you group them together rigidly (say with transport tubes), with half spinning in each direction, or if the outer radiation shell is spinning in the opposite direction from the inner shell, this is not a significant issue.