In Roman-Greek times (before 300 AD) there were no books in the Roman Empire, just papyrus scrolls, each chapter being one scroll. So a work like the Republic would be divided into many scrolls. Papyrus is relatively perishable, so virtually none of these scrolls survived, with only a few isolated exceptions.
The works have survived, however, because they were copied either by individual scholars or by being copied en masse in scriptoriums. In a scriptorium a master reader, called a lectorius, would recite the word and scribes would write it down as he spoke it. Another method of copying, more precise, was to use the punct, a sharp stick. With the left hand the copyist would place the point of the punct below the letter being copied and write it with the right hand. Then he would move the punct to the next letter and repeat the process. Our word "punctilious" comes from this process.
They began using parchment, the corium of a sheep, for the writing material, which is much more durable than papyrus. Note that the Romans did make some use of parchment, but the use expanded greatly in medieval times. Parchment was much, much more expensive than papyrus.
Every library in medieval times had a scriptorium. They would borrow scrolls or books, copy them for their library, then send them back.
At the Abbey of Cluny over 1000 ancient works survived by this method including those by Livy, Aristotle, Aesop, Horace and many others.