If you're willing to sacrifice some quality, it seems feasible to just download and store it locally. At 500 kbit/s for mediocre standard definition, it works

If you're willing to sacrifice some quality, it seems feasible to just download and store it locally. At 500 kbit/s for mediocre standard definition, it works out to about 200 MB per hour of video. At that rate a 4 terabyte drive would store ~20,000 hours or ~3 continuous years worth of video, which is probably several decades worth of typical YouTube viewing, assuming you watch the whole clip.