Could a paint made with nanites/nanotechnology be able to make, say a stadium, sound-proof just by painting the stadium with it?
No, and yes, but mostly no.
Traditional sound abatement walls rely on macro-features, you've seen the pictures of the silent rooms with the walls covered with triangular wedges. Although these rooms are eerily quiet, they are in no way sound-proof.
However, if you could make the exterior surface of a wall with active components consisting of trillions of microphones, and speakers and computing power, you could theoretically emit sound that just happens to cancel the sound, just as noise-cancelling headphones work. Again, noise-cancellation tech is not 100% effective.
Such a paint is not possible with current technology, but it is theoretically possible. It would require large of amounts of active components and power, but with nanotech in the limit, sure.
Unfortunately, not really though. Compared to nano-tech, and even the thickness of paint, sound waves are huge. At 440 Hz, the sound waves are around 70 cm wavelength. It is very difficult to cancel noise in general with components much smaller than the sound waves, just using speakers (in this case flat plate radiators) and nanotech would be far too small. Headphones have the advantage over a noise-cancelling painted wall, because they only need to do so in a very small area (directly over the ear, and can include simply sound insulation as part of the design) as well as being much larger than a layer of paint.
To actually cancel the noise using a very thin active surface, you would have to be able to suck and blow air (i.e., not just speaker vibration) for much longer periods of time than a microscopic speaker would be able to do. This would require some sub-surface plenum to capture or supply the air needed for the noise cancellation. There is no room for such a plenum in a thin layer of paint. So, no, unless the noise you needed to cancel was deep ultra-sonic (MHz or GHz perhaps), active paint systems won't really do the job. I am not even sure that such ultra-high frequency sound waves are even possible, but I am quite sure that they would dissipate very rapidly, obviating the value of our nano-paint.
Back to the big wedges on the wall -- there are the size they are because they must dampen the large sound waves in the room.
Active noise cancellation does not necessarily require digital processing, most headphones use analog processing IIRC, but some form of computing is necessary.
Related story: Many years ago I worked in a noise-control technology group for an engine manufacturer, and we discovered that some areas of an engine actually emitted negative sound power -- due the natural vibration modes in an engine, some parts would be out of phase with the dominant sources of engine noise, and actually absorbed sound due to wave cancellation. So we discovered noise cancellation (as others had in the past). The computing resources of the day (around 1980) were too expensive and large to use for such trivial matters as to make noise cancelling headphones, though we thought it was a pretty cool idea though. When I first saw products on the market, I thought, of course, computers are cheap enough now for noise cancellation.
I've spent a fair amount of time in anechoic chambers, prepping for engine tests, and it is kind of creepy. I would normally leave the big doors open to let outside noise in to avoid the creepy effect. The mass psychology of a stadium full of people in such an environment would be interesting.
The answer is: no.
In order to interact with sound waves you need something of the same order of magnitude as the wavelength.
The wavelength varies dramatically with pitch, but, as reference, the "diapason La" is 440Hz and its wavelength is 78cm - way above nanotech dimensions.
It is true that you could group together a huge group of nano-elements to reach the needed length, but it seems overstretching the technology for no added benefit.
This, by the way, is the reason why "cheap" anechoic chambers do not perform well on low pitches: their absorbing elements are not long enough.
Sound has many interesting properties, harmonics being one of the least intuitive. At its simplest, sound can be broken up into frequency and intensity, much as light is. Just like light, at higher and higher intensity, non-linear effects become more important. Your question pinged me into the realization that I really don't know what the maximum intensity theoretically possible is for sound (at STP) nor do I know what non-linear effects would occur as that limit was reached. I wonder what the maximum pressure difference is between peak and trough at a given frequency? I wonder what chemical and physical processes begin to be important sinks for acoustic energy as intensity increases? I mention this because "sound proof" without any qualifiers for frequency range and what the acceptable reduction in energy is, is fairly meaningless. (Most soundproofing both reduces intensity AND shifts frequency, it is not common for "soundproofing" to stop all acoustic transmission.)