This is the kind of experiment that comes from the theoretical concept of the problem, not from the engineering or scientific view of what we can feasibly measure or experiment with.
First, the angle you are trying to measure is incredibly small because Earth is very big indeed. The imperfections of the desk in front of you are by far greater than the imperfections that your string could tolerate. And the strength that the string should possess is greater than anything humanity has made. You would have to pull each end of the string with many millions of tons of force to make anything close to an acceptable measurement.
Look at simpler experiments: look at the paths the celestial objects (Sun, Moon, planets, stars) draw on the sky and check whether those paths are consistent with a hovering sky or with a round Earth. Look at whether the coordinates of places you know (that you can find with your own GPS) correspond to the "RE" maps, and correspond to a round Earth divided by parallels and meridians. All of the above has been done by the likes of Galileo, and by navigators all around the world for the last 500 years or more, even without the handy GPS signals.