ANC works best for low frequencies. As the frequency gets higher, hence wavelength is shorter, phase becomes more critical.
For example I own a pair of high end ANC heaphones (Sennheiser PXC-550) and their ANC mainly affects frequencies below, say, 1 KHz. For frequencies above 1 KHz the key is the good old mechanical isolation.
So, if voice is what really bothers you I would recommend good mechanical isolation headphones instead. ANC attenuates voice a bit, but I can hear voices around perfectly (albeit attenuated, because the lower end is reduced by the ANC).
ANC does wonders with airplane, train or bus cabin hum, the typical and annoying AC hum and other low frequency noises present in crowded buildings, etc.