I would like to find an electronic actuator that mimics the characteristics of a hydraulic actuator, in that the position remains fixed without power drain when the actuator is not moving. Which actuators exist that match these criteria?
Actuators that use a worm gear can't generally be back-driven; these are the same gearing mechanisms that let you twist a guitar string to tune it without the tension being able to back-drive (and thus un-tune) the tuning mechanism.
Worm gears are nice in that they offer generally pretty huge reductions in speed, with a huge increase in torque. This leads to them being used in lots of things, one of them being linear actuators. The worm gear will typically be used to step down the motor from high speed to high torque, which is then used to turn the leadscrew that operates the linear actuator.
So, if you're looking for an actuator that's electric that can't be back-driven, you can search for worm gear actuators, or just look for linear actuators as most of them will expressly state if the can be back-driven.
Another option is a motor which maintains position with a magnetic field.
Many stepper motors fall into a detent position when power is removed. If you only need whole step positional accuracy and the detent torque or cogging is large enough to hold your load in place, then this may be an option.
If the positional accuracy or detent torque is not high enough, then gearing can be applied to improve both at the expense of maximum speed (this is essentially what Chuck is suggesting, but a worm gear isn't the only option).
A latching relay remains fixed in its last position when power is disconnected.
A stepping switch is a specialized multi-position latching relay.