Making pasta with a pasta machine is one of those tasks that is extremely relaxing if time is not an issue and stressful if rushed. So do allow about two hours to prepare the fettuccine — it is worth it.
Adapted from "Recipes for Parties" by Michael Leva and Nancy Parker.
On a clean work surface, make a mound of flour and form a well in the middle. Crack the eggs into the well and lightly beat with a fork. Scoop a little flour from the outer edges of the mound and draw upward, allowing flour from the top to fall into the well. Using your fingertips, combine the flour with the eggs and mix well. Stretch the dough out and fold it back on itself. Add the oil and salt and knead for 5 minutes.
Cover the dough with plastic wrap and set aside for 10 minutes to soften. Divide the dough into 4 pieces. Work on one piece at a time. Cover the other pieces in plastic wrap.
Shape the pasta dough into a patty ¼-inch thick and feed through the pasta machine on the widest, thickest setting (gauge #1). Allow the dough to feed through on its own without forcing or pulling it. Once passed through, fold the dough in half and pass through again. Repeat this process 4 more times before switching the pasta machine to the next lower setting (gauge #2).
Feed through the dough before folding, and then repeat the process 4 more times. Switch the pasta machine to the next setting (gauge #3), fold the dough, and pass through 3 times. If the pasta becomes too long to handle, cut the dough in half widthwise and continue to process the halves individually. Pass the dough through the pasta rollers once on a lower setting (gauge #4) and again on gauge #5.
Lay the pasta sheets on a wire rack and repeat this process for each of the remaining pasta dough patties.
Change the rollers on the pasta machine to fettuccine cutters and pass each of the pasta sheets through one at a time. Lay the strands of pasta on a wire rack or hang on a wire coat hanger until ready to use.