In real life, I have over a meter of monorail and bellows, and a support rail...
¿Any problems?
This is all new to me, but I just had to do the math on this for the insane hilarity of the discussion: assuming you have 1 meter of steel rail, and there's a 1 degree (Centigrade) increase in ambient temperature during your shooting, that rail will expand 0.013mm. To re-iterate, this is 13 times more than the 0.001 accuracy you "need," rendering that level of accuracy entirely useless. With just 1 degree change in temperature.
So if you manage to get that stepping motor accurate to 0.001mm, be sure not to touch it, breath at it, or even get close to it during the shoot, as I'm sure the infrared radiation from your body will change the rail's temperature enough to make that level of accuracy irrelevant. Even if you wear an aluminum apron and operate the contraption from the adjacent room with a remote, heat generated by the camera electronics and friction heat from moving up and down the rail are another challenge.
Of course you could measure the temperature (passively) and take thermal expansion into account when moving the rail. But this assumes you have a perfectly rigid, perfectly straight rail bolted to a 1-ton granite slab in a cooled, temperature-controlled clean room built on top of bedrock way out in the sticks so passing traffic doesn't disturb it. Hell, I bet a a stray butterfly wing flap, a burrowing earthworm or moon phase would wreak havoc on the results
Source for thermal expansion calc. I used 13 as CLTE from
here.