Standard servo motors have limited range, but can be sent to any position and correct themselves. Continuous rotation "servos" interpret input as speed rather than location, but can turn repeatedly. Are there any servos out there that can turn repeatedly but can be controlled by location?
One example I know of is the LEGO NXT motors. They use optical encoders to be able to rotate fixed amounts without distance limits. Are there any other servos out there that can do this?
If the load isn't too high a stepper motor can be relied on to reach the desired position and number of rotations in either direction. An index mark (single slot optical interrupter) will allow you to detect if any steps are missed.
Any motor with accurate feedback can be use to implement a closed-loop positioning system.
You can add a rotary encoder to most motors. Just add money.
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You can add an external rotary encoder to almost any DC motor and/or gearbox. Some of them are designed to make this easy. For example I have a small DC motor with an integral epicyclic gearbox which has a stub shaft at the back to attach a rotary encoder. Using a conventional quadrature encoder and with a microcontroller providing positional feedback, this would produce something functionally equivalent to the NXT motor you mention.
What about these?
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Okay, I found what I need. (Thanks, Robin -- you were really close!) Pololu sells gearmotors with built-in encoders, the Arduino should be able to take care of the PID and such.