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.
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.
High pitch noise like from hard drives, typically in the 5 to 15k hz frequencies generally can be dampened quite a bit with pretty simple solutions like accoustic foam. Since it’s probably not that loud you won’t need much material to achieve a bit of reduction. Only thing I’d worry about is temperature management, keep an eye on it. Afaik there are also special desktop PC cases that are insulated, for like sound studios and such.