Doing it within the case would be a good science project but you'd have to locate the anti-noise speakers as close as possible to the noise source. You might try to find the location on the exterior of the case where the majority of the noise is emitted and locate your speakers there but that's not going to be less effective. Finding suitable speakers might be difficult and you'd have to be able to process the noise samples and emit the anti-noise quickly. Probably something best done by a dedicated DSP.
There are two ways to implement noise cancelling. The first way is to locate the anti-noise source very close to the noise source. This technique is sometimes used on jet engines. This won't completely eliminate the noise but it can lower it substantially. The second way is to locate the anti-noise source very close to the listener. This is, obviously, the technique used by noise-cancelling headphones.
If you locate the anti-noise source anywhere else, you'll have difficulty cancelling the noise everywhere in the room. In fact, I submit that it's not possible. Hand-waving example: Assume your noise source spits out a sine wave (sine waves are simple so let's use that). Now, to achieve perfect noise cancellation, you want your anti-noise source to emit a sine wave of the same frequency such that it's 180-degrees out of phase when it reaches the listener. This is simple: just ensure that your noise and anti-noise sources are in-sync and position the anti-noise source anywhere in the room but make it 1/2 wavelength closer to the listener than the distance between listener and the noise source. Adjust the amplitude of the anti-noise source to take into account the different distance. What you've done is to create a 'node' in the room where the waves cancel exactly. This demonstration is typically performed in high school physics classes. Now, add another listener somewhere else in the room. If the position of the new listener is such that the difference in distance to the noise source and to the anti-noise source is 1/2 wavelength, then that listener, too, will hear nothing. Chances are, though, that that's not the case and the noise won't be cancelled for the new listener. Worse, there will be some locations in the room where the noise and anti-noise waves are in phase so they'll actually amplify the original noise.
In the real world, it's more complicated since the noise emitted from your fans isn't a single frequency and you have to deal with sound reflecting off the walls, etc.