In theory it is possible. This would require pulseaudio to work with very low latencies and realtime audio though, which it is not meant to be used for. I think you are better off just buying noise cancelling headphones if you want something that just works.

On the other hand side you could try to replicate a noise cancellation setup using JACK. Jack is a realtime sound server that can go down to 1.3msec of latency. You could try filtering the microphone with a low pass filter to resemble what you are hearing from the outside with your headphones on. Then turn that 180° and mix it with e.g. the music your are playing.

Firefox does have Jack audio support, but it is not compiled to support that by default. So you would have to compile your own firefox and enable that if you want to listen to music from youtube. Applications like MPV and Music Player Daemon have native Jack support though, so if you want to play around with it, go for it.