People have been making paper for thousands of years, by hand or machine.
Turning growing plants into useful paper is somewhere between precise science and a black art, and frequently involves strange smells and odd stains, making it the original "mad science" (in fact, getting the secret of papermaking out of China was one of the earliest cases of industrial espionage...).
"Proper" papermaking equipment costs hundreds of pounds, but you can get perfectly acceptable results for under a fiver.
I have also posted an Instructable on actually making your own paper from scratch. Click here to read it