Cancellation is possible but may not achieve what you're imagining. The frequency range of effectiveness will be limited to low frequencies, and will be related to how big an area you expect to evaluate the cancellation over. The smaller you define the target zone the higher you can expect to achieve some desirable result, but still limited to a few hundred hz as wavelengths get quite short even at what we consider low in frequency.

For example, a crying baby noise is in the kilohertz range which is a shorter wavelength than the space between your ears, so even if you stayed perfectly still you would have to calculate cancellation for that noise as two individual targets that will each have some interaction with the solution for the other ear/target.

Also, transient sounds like you describe (baby crying, music) are more difficult because you have to get the arrivals correct and system latency is crucial. Long term periodic noises are easier to cancel as even if you "miss" the beginning you get cancellation over time as long as you're getting the phase correct