Earlier quoted context omitted.
Noise cancelling headphones will cancel low frequency sounds like fans / AC, but not high frequency ones like talking and phones.
Have you tried one? Mine significantly dampens and distorts speech so that it becomes easy to ignore. It's a recent Sony top-model, which from reviews seems to be on the top together with Bose regarding NC technology at the moment.
When they are on, but not playing music, I can still hear loud voices but it’s significantly dampened (like in the room next door).
With music playing, at a relatively low volume, I can’t hear anyone - unless someone is shouting right next to me.