PID control achieves zero steady state error with step set points and disturbances?
It's all about the integrator.  An integrator will cause the smallest error to increase without limit (at least until saturation, or until the control circuit drives the error to zero).  



So any difference between the reference and the feedback will cause the integrator output to change to the point where it drives the error to zero (and therefore there's no further change in the output of the integrator.)

With only a proportional gain there has to be some residual error to be multiplied by the gain in order to drive the error to as small a value as it can be, but it won't be zero.  A control system with a single integrator is type I: