Spherical mirrors for telescopes are not all that hard to make and can be made at home. By hand even! No cnc required. The mirrors are lapped using carborundum or diamond compounds. It can be done faster using a couple of gear motors though. After the mirror is ground you coat it with aluminum using evaporative deposition in a vacuum chamber. Scientific American has had articles in the Amateur Scientist section n the past on these subjects.
The reason they dont want you touching or cleaning the mirror is because the mirror is a first surface mirror where the aluminum reflective layer is deposited on top of the glass not under like a normal mirror. These surfaces are usually not hard coted so even cleaning can scratch or smudge the surface.