Solution 1: Your body makes precise chemicals all the time. Atoms' positions are predictable enough to position them very reliably. Atoms are much heavier than photons, so their positional uncertainty is much less. It was only recently that a very intricate physics experiment managed to show an interference pattern between sodium atoms.
Solution 2: One fertilized egg can "build" a whole human. Nanotech only requires a factory to be able to build another factory of roughly the same complexity.
Solution 3: Not all chemical products need to be buildable. If we can get just a few dozen reactions working, and build diamond in a variety of shapes, we'll be able to build self-reproducing factories; that's all we need for most of the amazing advantages I listed. (Food and biotech will require other reactions; however, those are already done by cells, so we know they're possible.)
Solution 4: This is an engineering problem, not a proof that it can't work at all. Energy can be delivered electrically, chemically (it can be converted in a space smaller than a mitochondrion) or mechanically (for example, with high-frequency sound waves). Nanotech will be much more efficient than today's machines, so it will be able to do at least as much work per volume without overheating. Also, the ability to build more intricate structures means we can use fractal cooling channels.
Solution 5: There are several proposals. My favorite involves many robots fastened down in a factory. The products of hundreds of robots are fastened together by a second-level robot; then hundreds of second-level products are fastened together by a third-level robot, and so on. Surprisingly, it doesn't take many levels to build a pound of product atom-by-atom. (Analysis may be found in _Nanosystems_ by Drexler.)