I'm neither a physics nor sound processing expert, but it seems like the microphone creates the signal to invert, and the speakers emit the inverted signal before the initial sound hits your ear. So I reckon the sound must first hit the microphone, then the speakers, then your ear. With headphones, the speakers cover your ears and the microphone is outside the enclosure, so that's easy. In a larger controlled environment with reliable sound dampening on 3 sides, maybe? My inexpert intuition says that doesn't stand a chance of working in an open environment with one microphone and two speakers usually closer to each other than the user's ears. It seems like having a door on a building with no walls.
In theory it should be possible with a standard sound card, or a sound chip in the motherboard.