[men8ifr]

Zero - your lack of knowledge is made up by your ambition but sadly I suspect your budget will not support the latter...

Also consider:

Steel thermal expansion is ~12 (um/(m K)

So for every 1 deg degree of temperature change you get 12 um in 1m or 6um in your 0.5m.

So options are:
1 - temperature controlled room/box with accuracy 1/6 of 1 degree - but even that has used up your entire 'budget' of tolerance... Plus you need to run the machine so slowly heat generated in ballscrews in negligible - here the problem lies - to remove backlash you need tight preloaded ballnuts... tight ballnuts generate temperature.. so with very tight ballnuts I would expect a speed of a small number of mm/min to avoid any temperature effects....
To be honest none of this is feasible.
2 - Run closed loop control to 1um scales using kflop, mach servo drives (Australian look on ebay) or linux CNC. But factor in 1um scales accuracy (if true) will be +-0.5um so, again that's all of your required tolerance used up in the scales and I bet movement of the workpiece relative to laser is not identical to the scales. also I would only trust a reputable brand e.g. mitoyoto and you need to study what their accuracy is, I doubt a 1um glass scale means it is within 1um everywhere and again I believe these are temperature sensitive so you need to understand how their measurement is changing with temperature.

[men8ifr]

Sorry just read Jim's post - you can add Galil to the options of closed loop control (and probably the best but $$$)