We made
our own "string potentiometer" for 2015 from a tape measure and a 10-turn pot from Jameco.
A roller as suggested by Ernst is a similar concept; make sure you know how many total turns it will make as the slide travels, and leave some spare for slippage.
Linear slide potentiometers are also available, mostly for much shorter lengths than the string unit.
An optical encoder can also be used directly on the linear slide. Use an LED and sensor module like you would use for tracking a line, then draw lines on the slide perpendicular to the direction of motion*, and count how often the surface becomes light and dark. You can even use two of these and make a quadrature encoder to keep track of the direction of motion.
If your counter is subject to slippage or loss of data due to missed transitions, you may want to add at least one limit switch to the slide to allow for an automated calibration.
* or print lines on a long label and stick it on the slide
Edit, if you don't mind doing way more processing than should be necessary, you can do what inspired our electric tape measure: Fix the loose end of the tape measure to the slide, and the housing to the fixed part of your device. Then, point a camera at the tape near the housing, and do a little OCR to read the tape measure. Probably not practical, but it led to a better idea.
Edit2: electronic calipers already do what you want to do - see if you can find some which will allow you to take a digital reading rather than drive a display.