I'm attempting to build a machine that scans the surface of an object and measures the various heights of the surface. The output from the laser is an ethernet connection. I plan to use MATLAB to both control the machine and process the resulting data. Is something like this even possible? I know I'll need various shields (relay for stepper motors, ethernet, etc.)
Yes this is possible.
But what exactly do you want to know?
Also can you explain why you want to use an ethernet connection and MATLAB for control ?
Try searching for LIDAR arduino modules.
See this LIDAR scanner project (http://www.eevblog.com/forum/projects/hobby-lidar-build/).
Video of it in operation: https://youtu.be/0IQDjnow9k8 (https://youtu.be/0IQDjnow9k8)