[JabrZer0]

This particular technique has its problems, as has been mentioned by other people. I would think inverting and cancelling is much more useful in situations where you can isolate the background noise in real time. For example, I do a little hobby music production, and I've heard of this being used to isolate vocals:

Because the vocals are sometimes the only thing that's perfectly in the middle of the mix (balanced left/right), you can invert the left channel of a mix and then recombine it with the right to cancel out only the balanced tracks (so you're left with essentially an instrumental). Subtracting the instrumental track you've just created from the full song (again, by inverting and recombining), you should get the vocal track by itself. Fun stuff, right?

I'm not sure how well this works, as I've never done it myself, but it would seem like having the exact signal is important. As people have said, the fan noise isn't predictable enough to cancel using that technique. On the other hand...

I recently read about a machine learning application using K-means clustering in which you feed the algorithm sound from two microphones set up in the same room and it's able to separate the audio by who is speaking. I'm not sure how well such an algorithm would work when one sound is significantly louder than the other (such as quiet fans and people talking), but it certainly points to that being a possibility.

[frank_jaeger]

As a part time MC, I can attest that inverting and recombining channels of a song can occasionally lead to a decent instrumental. That being said, with little talent or background, and using freeware, it will not be perfect. It will be much better than using the original track, but it will certainly be noticeable that you are not using the instrumental. Great for practice though.

[dharma1]

NMF and ICA can also be used for blind source separation, but there are artifacts. Good for some applications like speech recognition, but for music you really want no audible artifacts