play --show-progress -c 2 --null synth 01:00 brownnoise \
    reverb 19 vol 0dB bass 6 treble -3 repeat 558

Adjust bass and treble to match your headphones.

It sounds like a big waterfall, or like a continuous wave crash. And it's stereo. :)

EDIT: Or this:

    play --show-progress -c 2 --null synth brownnoise \
    reverb bass 6 treble -3 echos 0.8 0.9 1000 0.3 1800 0.25

The one with echos is a bit softer and more natural-sounding.

Stupid question: Is it necessary to play something through a pair of noise canceling headphones if you just want to "cancel noise" with them?

Not with the ones I'm familiar with (Bose QC-15). You have to turn the headphones on to engage the 'active' noise cancellation, but you don't have to be listening to anything. In fact, you can unplug the cable from the ear piece completely and still hear (not hear?) the effect.