Is there any way to obtain linear or angular distance proportional to a voltage signal?
"Servo" is exactly what you're looking for. It's a generic term for mechanical movement that's proportional to an electronic signal. Digital servos typically use PWM control signals, but old-fashioned analog servos use voltages as control signals.

The general idea is that you have an actuator, typically a DC motor or solenoid (AKA "voice coil") and some means of measuring its position. You compare the measured position to the control value, and drive the actuator based on the error signal.