> 0. Be not a high drain on power.
The HC-SRO4 that I have seems to draw about 5 or 6 mA. I power it from a digital output pin so I can turn it off if necessary.
> 1. Be reasonably priced.
HC-SRO4 ultrasonic distance sensors can be had for under $2 each: http://www.fasttech.com/product/1012007
> 2. Be omnidirectional.
I don't know of any such sensors. Perhaps you can use a bunch of somewhat directional sensors.
> 3. Be reasonably accurate in measuring the distance. Since I am not using it in a robot for object avoidance, I can tolerate a little bit of inaccuracy.
The ultrasonic sensors are reasonably accurate.
4. Responds only to human proximity. I don't want it to respond to non-animate objects, like closing doors etc.
Good luck with that. I don't see how you can measure distance to an object only if that object is attached to a human. If you could tag your girlfriend with a Bluetooth Low Power tag you might get a rough proximity reading. That option might not be available on Arduino for a reasonable price.