Other challenges with accuracy are:
Room issues: floor rigidity and stability (in addition to temp). A regular concrete floor may move over time. Large CMMs aren't just machines, they are installations -- same for CNCs.
Component quality, which translates as $$, good decisions, and experienced attention during installation.
Alignment: linear (easiest but not easy), 2D (harder but there are techniques), 3D (hard), and how to maintain alignment over time (hard if the machine is not massively built out of proper materials). In 1D, for example, linear rails are not straight when you buy them, so we can't rely on them for linearity. The usual technique is to machine a highly accurate registration ledge the full length (into a very stiff and stable frame), and force the linear rail against the ledge. Next we need to get the other rail extremely parallel to this master rail, mount it there, and have some way to ensure it stays parallel (what if you move the machine?). In 2D and 3D it gets more complex.
Dimensional stability over time: will residual stresses relax and cause parts to shift over time?
Rigidity, vibration, and damping due to cutting forces, inertial loading, and local forces: FEA is good for this, but accurate modelling of a CNC machine is not simple. Some things can be certainly calculated with equations and analytic methods, e.g., how bearing stiffness and slop geometrically propagates to the cutter tip. For the complex geometry, structure, and loading of a CNC router, only experienced use of FEA will do a good job of predicting stiffness at the cutter, or vibrations of the system. The consensus view of experienced machinists is that it takes massive amounts of cast iron and $ to achieve high accuracy over a large distance.
Machine design/configuration: CNC routers are complex machines with many issues. Assuming someone already has a mechanical engineering degree and many years of machining experience, I'd guess it would take about a year of CNC-specific study, *from the right people*, to be qualified to design one that could hold 0.001" 3D accuracy over a long distance, for years. I'm an experienced ME, have done design/build projects for 35 years, have been studying CNC design on and off for 5 years, and I'm not sure that I could do it (nor do I have much interest in it... seems like a bunch of headaches!) This kind of accuracy is just very difficult and complex.
So hopefully you can see why 0.001" accuracy over a long distance will cost an amazing amount of $, and more importantly, a team with lots of knowledge and experience.
For a point of reference, here is Multicam's top router, I'm guessing about a $70k machine:
7000 Series CNC Router - CNC Cutting Machines for Your Application & Budget | MultiCam
They only spec a repeatability of +/- 0.001". No accuracy is stated because it's considerably worse than that. But few people need 0.001" accuracy over a long distance.