[Al Skierkiewicz]

phrontist,
Yes it is possible for you to build noise cancelling headphones. What makes them expensive is all of the auto adaptive circuitry reguired for varying acoustic conditions. Effectively what you need to do is sample the noise, phase shift it so it arrives at your ear 180 degrees out of phase with the noise and at the same level and there you are. The auto circuitry samples the environment, determines what is noise and only amplifies/shifts that part of the spectrum with a little AGC thrown in for level adjusting. With DSP on the rise, it is really easy to get a good sample, analyze what is audio and correct for the rest. If you have a good idea of what the spectrum of noise is, then you can build a simple filter to just pass the noise to the phase shifter and amp. Throw in a tuning control and level control (knobs on the box) and you don't need to add the auto circuitry. Just tune out the noise as best you can.
So here is what you need...Mic preamp, tuneable filter, phase shifter, power amp. You should be able to put it together with parts from Digikey into a case that fits into your pocket. LM324 opamps should do nicely for the everything but the power amp. LM386 for power amps (you need two for stereo phones.) Have fun! There is a simple book "Op Amp Cookbook" that ought to be a big help.