Line a large colander or sieve with a thin, clean dishtowel (linen or cotton, not terrycloth, works best). Line a tall bucket with a clean plastic garbage bag, and then place a large bowl at the bottom of the bucket inside the bag to catch the juice. Rest the cloth-lined strainer on top of the bucket. (Alternatively, place a rack over a large bowl with the cloth-lined strainer on top. The point is to leave room to collect as much juice in the bowl as possible.)
Quarter and core the apples, leaving the skins intact. Chop the apples, then, working in batches, puree the apples in a food processor with 1/3 cup water per batch. Blend for 1 minute, scrape down the sides, then blend for 1 minute longer. The apples should be completely pulverized.
Pour each batch of apple puree (or mash) into the lined strainer. Allow the cider to drip through to the bowl beneath, stirring and pressing on the mash periodically to help release as much cider as possible. Continue with this process until all of the apples have been juiced. (The cider will darken as you work.)
Bring together all four corners of the dishtowel and twist into a hard ball to squeeze out any remaining cider. (Note: if the bowl fills during juicing, pour the cider into another holding vessel. A second dishtowel may also come in handy if you yield an excess of mash.) Unwrap the towel, stir the mash, and then twist and squeeze again a couple more times to express all the cider.
Refrigerate the apple cider immediately and drink it fresh within five days. Or, to keep the cider longer, boil it over high heat to 160°F to pasteurize it. Store for up to two weeks in an airtight container in the refrigerator. For longer-term storage, freeze the cider, being careful to leave 1/2-inch of headspace in each container to allow for expansion of the liquid. This will preserve the cider for up to 1 year.