No, you don’t need a pasta machine to make pasta. A pasta machine does make the process easier, but it isn’t a necessity.
What the pasta maker does is make it really easy to roll the dough really thin, and make it easier to cut with attachments. However, while a little bit slower, this can be done by hand.
Simply roll out the pasta dough into thin sheets with a rolling pin or other heavy round utensil, and use a knife to cut the pasta into your desired shape.