Italians use a very fine flour called "00" in pasta dough, but all-purpose works well, too. You won't use all 4 cups of flour, but the extra helps keep the eggs contained in the well.
Dump the flour in a pile on a work surface. Make a deep, wide well in the center and pour in the eggs, olive oil, and salt. Begin mixing the eggs with a fork, staying in the center and being careful that the eggs don't breach the wall. Little by little, mix in flour from the sides until the dough starts to move as a unit and is too stiff to mix with a fork. Continue mixing by hand, incorporating more flour to stiffen the dough. When it doesn't easily absorb more flour (one signal is floury, dried bits of dough flaking off the mass), set the dough aside; scrape up all the remaining flour and pass it through a sieve to sift out any dried-up bits. Discard the bits and keep the cup or two of sifted flour on the work surface to use during kneading if necessary.
Wash and dry your hands. Knead the dough on the lightly floured surface until it's a smooth, homogenous ball of dough, firm but resilient, neither too dry nor too soft, about 5 minutes; it should no longer stick to the surface. Poke it and it should spring right back; press your finger into the center and it should feel just a bit tacky. If it's very sticky, knead in more flour.
Wrap the dough loosely in plastic and refrigerate for at least 1 hour, or up to 8 hours.
Set up a pasta machine on your work surface. Divide the dough into four equal pieces. Work with one piece at a time, keeping the others refrigerated, wrapped in plastic. Flatten the piece of dough with your hand (flour it lightly with necessary) and run it through the widest setting on the machine twice. The first pass roughs up the dough; the second pass smoothes it out.
Set the rollers to the next narrower setting. Pass the dough through twice. Continue notching down by one setting and passing the dough through two times. As the dough lengthens and thins, it will bunch up under the machine. Rectify this by gently lifting it out and folding it neatly behind the machine.
When the pasta is 1/16 inch thick (you probably won't need to go to the narrowest setting), lay it on the work surface and cut it into 12-inch rectangles. Using either the tagliatelle cutter on the machine or a knife, cut the pasta into 1/4-inch strips. The tagliatelle can be dried on a drying rack or cooked immediately.
Photo: Scott Phillips