In a medium skillet over medium heat, cook spinach, stirring occasionally, until wilted, 3 to 5 minutes. Remove from pan, and cool. Squeeze out excess liquid; there should be 1/4 packed cup of spinach.
In the bowl of a food processor fitted with the steel blade, puree spinach until fine. Add 1 egg, and process few seconds more. Set aside.
Mound flour in center of a work surface, and make a well in middle. Crack remaining 3 eggs into well. Beat eggs with a fork until smooth, then add spinach mixture to well and stir until combined. Begin to work flour into mixture with fork. Use a bench scraper to work in the remaining flour, a bit at a time.
Once all the flour has been incorporated, start working dough with hands to form a rounded mass for kneading. Be sure the work surface is clean of all loose bits of dough; lightly dust with flour. Knead dough until smooth and elastic, about 5 minutes. Cover dough with an inverted bowl or plastic wrap, and let rest 1 1/2 hours at room temperature, or refrigerate overnight.
Divide dough into eight pieces. Working with a few pieces at a time and keeping remaining pieces covered, flatten each portion of dough into a disk shape somewhat narrower than machine opening. Very lightly dust dough with flour. Feed through at pasta machine's widest setting, keeping in mind that machine settings differ; some have as many as ten, while others have only six. (If dough pulls or tears when passing through machine, sprinkle a little more flour over dough before feeding it in to keep it from sticking. When you're finished, remove excess flour with a dry brush.) As pasta sheet emerges, gently support it with palm, and guide it onto work surface. Fold sheet in half. Pass through a second time on the same setting to smooth dough and increase its elasticity.
Thin dough by passing it through ever-finer settings, two passes on each setting, until sheets are almost translucent but do not tear.