By just taking standard equations found in textbooks, plugging in some numbers, and realizing that things such as Ideal Gas Law don't hold at such scales.
You can even derive the equations for yourself to confirm your not being misled by the textbooks, but that takes more skill and knowledge then is likely feasible to acquire without advanced study.
We will face different challenges at different scales, of course, but even as recently as 2016, researchers using a scanning tunneling microscope crafted a 8,192 bit message using less than ten thousand atoms. So, no, I see absolutely no reason why we couldn't have robots within an order of magnitude of the human cell -- we have bacteria which are much much smaller capable of a variety of functions as well as reproducing themselves
You might look into Feynman's "There's Plenty of Room at the Bottom."
Clearly cell-like robots are 100% possible at very small scales, there’s no one debating you on this.