In a food processor combine the flour, salt, eggs, oil and 4 tablespoons water and process until the mixture begins to form a ball, adding more water, 1 teaspoon at a time, if the dough is too dry. Process 30 seconds more to knead it. Remove the dough from the processor and let it rest, covered with an inverted bowl, at room temperature for 1 hour.
Set the smooth rollers of a pasta machine at the highest number. (The rollers will be wide apart.) Divide the dough into 4 pieces, flatten 1 piece into a rectangle, and cover the remaining pieces with an inverted bowl. Dust the rectangle with flour and feed it through the rollers. Fold the rectangle in half and feed it through the rollers 6-8 more times, folding in half each time and dusting with flour if necessary to keep it from sticking. Turn the dial down one notch and feed the dough through the rollers without folding. Continue to feed the dough through the rollers without folding, turning the dial lower one notch each time, until the lowest notch is reached. Roll the remaining pieces of pasta dough in the same manner.
Use the blades of a pasta machine that will cut 1/4-inch wide strips. Feed one end of a sheet of pasta dough through the blades, holding the other end straight up from the machine. Catch the strips from underneath the machine before the sheet goes completely through the rollers and put the cut strips lightly across floured jelly-roll pans or let them hang over the top of straight-backed chairs or on hangars. Let the pasta dry for 5 minutes, before cooking.