Wow, you guys are hard core. I wouldn't want to do it either. I have backpacked many times and we always came back to camp. Guess what? There were showers! They weren't the best but they worked.
That is day hiking from a base camp (and apparently a drive in camp at that). Backpacking is carrying your shelter/bed and everything else you need several miles into the backcountry, where there are no facilities of any kind. In the morning, you wake up, pack up, and hike some more.
It is possible to stay hygenic, if not exactly clean. A few baby wipes, some hand sanitizer, and something to pull your hair back are the basics. I've known backpackers that skip toothpaste, but that's carrying it a bit too far for me. When you start feeling really grimey, a tiny wash cloth, a couple cups of warm water, and a few drops of Dr Bronners camp soap can make a real difference. It's just a few days, and the pay off is totally worth it.