It certainly does sound doable, at that price. We actually built a "tape measure potentiometer" for our robot with a 73-inch capacity for under $30 using COTS parts and a drill press. This cost did not include the RoboRIO, but it could just as easily have been an arduino. As you can see in the attached photos, we used a voltmeter to get analog readings before we ever hooked it to the 'RIO. If you use a 10-turn potentiometer (the costs go up considerably for higher number of turns), you'll need a drum/reel with at least a 30cm circumference, or about 10cm in diameter.
With the low retraction force, you'll need to may wind up with some "bird-nesting" on the reel if the load moves quickly towards the sensor.
OBTW, there is already a COTS item on AndyMark which looks similar to what you want, though it only has about a tenth of the extension:
http://www.andymark.com/product-p/am-2674.htm