Yes. Well, I don't know how "normal" the faucet would be, but you can get water to a tap without electricity.
I live in a small city. Even when the water goes out, we have water, because the city water is all gravity fed. On a hill, above all the houses, is the water tower. Except it isn't really a water tower, it's a gigantic cistern on a hill. The filtered, treated water gets put in the cistern (rather than collected rainwater, for example) via something called the
Venturi effect.
This is a very, very old system. This is what they used to get running water in ancient Greece and Rome, and in castles. The major difference between this system and current sewage systems is that we have flush toilets now. In the old days, they separated greywater from human waste, and the two were dealt with very differently. Now, they all go to the same place.
On a small, cabin-in-the-woods scale, if it were me, I'd have a big cistern with some type of lid, probably on legs, higher up than my house. I'd have it hooked up so that it passed through a filter (or filters) first, as well as some type of hot water heater (solar for summer, wood for winter, propane for backup, probably). I'd take the lid off to let it fill with rain, and put the lid on to protect it from contaminants once it was full. Depending on the climate, I might fill it with snow during the winter.
You can also move water with a
non-electric pump that's powered by a windmill. Have you ever seen those accordion-looking hand pumps that you use to pump a stability ball? Or an old fashioned fireplace bellows? Those are non-electric pumps. If you can picture it, a big giant one of these in/on a well, being pushed- well, pumped- by the motion of the spinning windmill. The reason that the Netherlands is known for windmills isn't because of grain (which, technically, is what a windMILL actually does, uses wind to move giant stones to mill grain into flour), it is because the Netherlands is one of the European low countries- it's swampy. So the Dutch built windpumps to keep water out of buildings so they'd last.
