Put ¼ cup of water into a large pot and bring to a boil over medium-high heat. Add spinach and cook until completely wilted, about 3 minutes. Drain in a colander and, when cool enough to handle, squeeze out water. Finely chop spinach and set aside.
Sift flour into a mound on a clean surface. Use your hand to make a well in the center. Add spinach to well, then add eggs and lightly beat together with a fork. Continue beating, gradually incorporating flour from the inside edge of the well into spinach mixture. When dough becomes too stiff to work with the fork, knead flour into spinach mixture with both hands until dough is no longer sticky. Form dough into a ball and cover with a damp kitchen towel. Clean work surface, then dust with more flour. Uncover dough and knead with the heel of your hand until smooth, about 5 minutes. Cut dough into 4 pieces and cover with a damp kitchen towel.
Working with 1 piece of dough at a time, flatten dough so it will fit through the rollers of a hand-cranked pasta machine. Set rollers on widest setting, then feed pasta through rollers 3 or 4 times, folding and turning pasta until it is smooth and the width of the machine. Decrease setting by 1 notch and feed narrow end of the pasta through rollers. Repeat, decreasing setting by 1 notch each time until you get to last notch. (Don't roll pasta through last setting or it will be too thin.) Cover pasta sheets with damp kitchen towels to keep from drying out until ready to use in Baked Spinach Lasagne, or cut into other pasta shapes.