Squeeze excess moisture from spinach. Pulse spinach, flour and salt in a food processor until well combined.
Beat eggs in a glass measuring cup with a fork until blended. With the processor running, slowly pour eggs through the feed tube. Stop, then pulse until mixture resembles coarse meal, about 1 minute (when pinched, the dough should cling together). If the mixture seems too dry, add water, 1 teaspoon at a time; if it seems too wet, add flour, 1 teaspoon at a time, pulsing between additions.
Turn the dough out onto a work surface and knead into a smooth ball. Wrap in plastic wrap and refrigerate for 30 minutes.
Cut the pasta dough into 12 equal pieces; cover all but one piece with plastic wrap. Flatten the unwrapped dough with the palm of your hand and pass it through the roller of a pasta machine turned to its widest setting. Fold the dough in half and pass it through the machine again, repeating 6 to 8 times, until the edges are smooth.
Pass the dough through successively smaller settings until it is the thickness of a dime. Place on a kitchen towel and let dry for 5 minutes. (Pasta dough is very sensitive to humidity. It should be firm and supple. If it seems sticky, dust with flour.)
To make fettuccine or tagliatelle, feed sheets of dough through the ¼-inch cutter of the pasta machine.
To make linguine, feed sheets of dough through the ⅛-inch cutter of the pasta machine.
Repeat with remaining dough. (Pasta that is not being cooked immediately may be spread on a baking sheet, between sheets of flour-dusted parchment paper, and refrigerated for up to 6 to 8 hours; freeze for longer storage.)
Make Ahead Tip: Refrigerate well wrapped for up to 2 days.