I have an idea using a servo and an arduino with keypad where I could punch in the value of a resistor and have a "barrel" spin to the right place, and the drawer open to said component. How feasible is this?
By barrel, I presume you mean a 'lazy Susan' (rotating table) type layout with several layers of drawers to hold all the different values of resistor. You'll need a servo for each layer of drawers or one servo on a linear actuator to lift it to the required layer. You'll also need a stepper motor to turn the table so that the required drawer is in front of the servo (and in front of you!).
Have you thought about how the drawers will be closed after use? Manually or by the servo? If by the servo, you'll need some method of attaching the servo to the drawer so that it can pull it shut and then releasing so that it can move to the next drawer.
Your input requirements will depend on whether the equipment can be directly connected to a PC and you make your selection using the PC keyboard or can be wirelessly controlled by a PC or you want it to be totally self contained with its own keypad.
The resistor values/ drawer placements can be held in an array. Look up the value in the array and the position will be in the array at the same place. EG. 150K27 =150KOhms are in layer 2, drawer 7.