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Noise cancellation relies on precisely controlling the distances to listener. If OP were to simply set up a tone generator near the hard drive, the waves would alternately constructively and destructively interfere as OP walked around the room.
That's not entirely true, cancelling sound waves can also be done on a large scale. I've seen it on festivals or other open air venues, where they stack two rows of subwoofers behind each other in a precisely calculated distance to cancel out the bass notes towards the back of the stage, meaning walking in front of the subs you could hear it normally, walking behind them and the bass notes where very faint and clearly coming as a bounce back from walls or the like. So they effectively cancelled out the frequencies towards the back.
But that doesn't help OP since that works way better on lower frequencies, with higher notes you'll need the precise localization you're talking about. So the high pitched noises from a hard drive probably won't work.
Given that they're both coming from the same laptop, I'm not so sure.
With low frequencies, where it is hard to locate the source of the sound anyways, it works well, even without carefully arranging the distances, higher frequencies get trickier and you'll likely get good cancellation in some places and a terrible mushy noise in others.