Would I be able to feel a single-molecule wide string? Imagine there is a material that is so strong it can form several meters string while being only a single molecule wide. It is securely attached and can withstand any human pressure. What happens if I slide my finger across it? Would it cut it or would I not notice anything?
 A: the idea of a single-molecule "fiber" of macroscopic length (called a "sinclair molecule chain") appeared in the science-fiction stories of Larry Niven some 50 years ago, an abbrievated analysis follows. 
If we imagine a string consisting of a single linear molecule, that string would possess no dislocations, point defects, or grain boundaries, and its breaking strength would not be limited by the (relatively weak) forces that cause adjacent polymer chain molecules to adhere to each other in things like spider web silk. 
As such, its breaking strength would be equal to the load required to actually break the chemical bond between any two atoms comprising the string's backbone, which would be many orders of magnitude greater than the load required to cause two of those chains (of shorter length) to slip past one another, or for dislocation travel (in a macroscopic fiber) to promote plastic deformation, grain boundary rotation, or the merging of point defects. 
Niven invented the single-molecule chain to furnish an easily-concealed tool in the form of a long molecular chain with the cross-sectional area of one molecule and knobs on its ends which a person could unfurl and then use to cut through any solid with little effort. However, he didn't do the math- which of course had no effect on the entertainment value of the sci-fi story, but a reader who happened to be materials scientist would reject the premise. 
A "real" polymer rope of finite cross-section might possess in cross-section somewhere around 10^20 polymer chains and have a breaking strength (based on slippage between adjacent chains) of order ~tens of thousands of pounds. Niven's premise was that the tensile strength of one molecule was of order ~10^20 greater than the strength of a collection of those molecules tangled up against one another in a solid, so that a single molecule string would possess the interatomic breaking strength of  ~10^20 tangled molecules held together with Van der Waals forces. 
But what he failed to recognize was that a load of several hundred pounds applied to the ends of that single molecule string would produce a stress of ~ (load)/(cross-sectional area) and the area involved would be that of one molecule or more properly the cross-sectional area of the electron orbital overlap which furnished the interatomic bond. 
With such a tiny number in the denominator of that fraction, the stress would be almost unimaginably gigantic, and pulling on it with your hands would easily break it.
A: It would not cut you, not any more than a knife could cut through a bucket of water. And at scales this small, you're basically indistinguishable from a sack of water with some impurities.
The mechanoreceptors in your skin sense pressure and motion on the scale of entire cells, so they wouldn't be tripped. That means you wouldn't feel anything.
It may cause some damage to macromolecules such as DNA by cutting through them, but a sufficiently small number of cells would be damaged so that you wouldn't suffer any ill effects, just like you don't when you get an X-ray; it would be negligible compared to the DNA damage you spontaneously get. In fact, a good analogy is cosmic rays. Millions of cosmic rays will pass through your body every day, acting much like your ideal string. This is not a health problem, though it is for astronauts (who get much more of them), possibly increasing their risk of cancer.
