In mweiss' method, the outer angles ∠AOC and ∠DOB are shy of 1° by slightly more than 0.0001°, and the inner angle ∠COD exceeds 1° by slightly more than 0.0002°.
On a (huge!) one meter diameter protractor, that is a worst case error of less than 4 microns, or one-seventh of a thousandth of an inch.
Formulas used to calculate the "shy-ness" of the outer angles:
α = angle being trisected
inner angle = 2 * arctan(tan(α/2)/3)
shy-ness = arctan(tan(α/2)/3) - α / 6
If that accuracy is not good enough for you, you are welcome to repeat the process:
- Mirror image ∠DOB about the line OD,
- to make an angle ∠AOE that exceeds 1° by slightly more than 0.0002°.
- Angle ∠COE is slightly more than 0.0003°.
- Use mweiss' method to approximately trisect angle ∠COE.
- Let "F" be the new point that is one-third of the way from C to E.
- Angle ∠AOF is our second approximation of a degree.
Using the previous formula, and impossibly accurate compasses and straight-edges, angle ∠AOF is shy of 1° by slightly more than 10-16 degrees.
On a protractor the size of the Earth, 90° would be 10,000 kilometers. Even at that scale, angle ∠AOF would only be shy of 1° by slightly more than 10-11 meters.
In our quantum-mechanical world, you cannot draw anything that accurately.
"Solid" matter made of nuclei and electrons gets really fuzzy at a scale of 10-10 meters. If you try measuring any smaller than this scale, you cannot know whether an electron will be where you are prodding or not. So this second approximation of a degree is limited by Earthly physics: you cannot make a compass or a straight edge perfect enough to achieve it, and even if you could, you could not measure how perfectly you drew it.