Look, I know, TV shows and even some headphone commercials have made this a little confusing. You know – marketing wankery.
So I’ll explain what Active Noise Cancelling (ANC) actually does. ANC uses microphones outside each speaker housing to “listen” for predictably repeatable sound patterns. Things like fans, engines, motors, condenser pumps, tire treads on the highway. Any sound that is a rapidly repeating pattern, that they can counteract with an opposite-phase rapidly repeating pattern.
People saying things, annoying or otherwise, is not the sort of droning, rapidly repeating pattern that ANC helps with. If you’re trying to block out the chit-chat, just regular headphones/earbuds are all you need.
Anything that isolates your ear canals from a direct path from sounds in the environment will have a great effect on reducing that sound. Think about earplugs that you might use at a shooting range or industrial site. Earplugs don’t have batteries. Don’t make sound. They just plug the holes. Now think about hearing protection that have a head strap/spring and cups over your ears. Passive. They just isolate and block. Very effective. Nothing active about them.
Therefore, regular headphones/earbuds, plus music (or something) is all it takes to block out the co-workers’ inane talking. ANC is an unnecessary (irrelevant) complication/expense.
Now – here’s the thing that’ll knock you over, if you haven’t understood it yet. ANC can make it easier to hear other people talking, especially if they’re talking loudly to be heard over the droning sound of machinery. They do this with hearing aids. Suppress the droning machinery noise with ANC, pass (& amplify) speech.
Every time some dimwit says “noise cancelling” when their intent is to block other people’s voices, a cute little fluffy kitten goes deaf.