In this paper, a simplified mathematical model of a telescope system is established, and a linear active disturbance rejection controller (ADRC) is implemented in high precision position and velocity control of a telescope servo system. The internal mechanical resonance and some nonlinearity in the tracking system, in addition to external disturbances of uncertain characteristics are collectively treated as a generic disturbance, and this total disturbance is estimated by a proposed linear extended state observer. This total disturbance is then rejected by disturbance compensation. The control algorithm employs only three parameters, which makes the controller very easy to tune. Experimental results are presented to demonstrate the good performance of the proposed telescope tracking method.