accurately move an object in linear steps with electro motor
You use a linear slide. These are available ready made in a huge range of sizes, load bearing and speeds.



Figure 1. A random image selection from Newmark Systems shows a screw driven slide.


Generally your required speed and load will determine whether to go for a belt drive (fast) or screw (high load).
You will then need to decide on a drive system - typically stepper or servo. Stepper is simple and cheap and good enough for many applications provided they are unlikely to stall. Servo uses encoder feedback so it can detect and signal positioning errors.


If you have a simple sequence of steps you may find a programmable controller to do that automatically. Typically these will have an LCD and keypad to allow you to enter preset stop positions which are selected by digital (on/off) inputs to the device. Others will be programmed by PC but still use the digital inputs. Digital outputs will signal busy/ready and fault.

There are also many DIY linear slide mechanisms used on CNC, 3D printers, etc.

Regarding your high accuracy requirement you need to watch out for backlash (typically a little play in the drive) and the effect of step size if using stepper drive.

Finally, you are going to have to make some power calculations based on the required acceleration and deceleration. (Your application will be simple as the load will not be accelerated by gravity.) Many of the slide manufacturers will have online tools for this. Have a look at (another random link) Why RMS torque is important for motor sizing to get you in the mood.