For a model crane with 3 axes (actually, I would consider 4 axes -- see later), I would consider using servos as are used in model airplanes, cars, etc. By using a servo, you largely avoid the need for position sensors, as that function is built in through the feedback from the servo potentiometer.
Get old, cheap servos rather than the more modern, highly integrated ones. You will need to hack the servos, and it is easier to do that with older ones that have discrete components.
Most servos will have limited travel (such as +/- 30 degrees). That is limited by the feedback potentiometer in the case and by a physical stop. You can remove the physical stop (in many, but not all) and pull the leads for the potentiometer out of the case. Now, you have a sensor (your external pot) and a motor that will turn to a controlled position depending on the pulse width of the signal you send to it and the postion of the pot shaft. For example, if you need 360 degree rotation, you can use a 360-degree or multi-turn pot connected to the item that is turning and rotate to any position in that range.
For linear motion, such as up/down, used a loop with dial cord and have the servo drive the loop and the pot driven by the loop (i.e., a capstan arrangement).
The microcontroller can produce the pulse width you need for each servo.
Now, for the 4th axis. As has been sugggested, full size cranes often ride on RR tracks. The crane can rotate, lift, and go back and forth on the track (3 axises). However, the up and down mechanism can also go back and forth on the lifting boom (i.e., closer to and further away from the center of rotation). In other words, like a construction crane. That in-and-out motion is the 4th axis to which I referred.
There are no doubt lots of resources on the internet to help you with the mechanical design. Check out robotics, RC model ship building, RC model trains, and RC model tanks. John