Is it illegal to make a famous book into a play without buying the legal rights?

If the book is still protected by copyright law, you will need permission from the copyright owner to adapt it into a play. However, if the book is out of copyright or the book's subject matter is universal, you can create the play without permission. For example, the theme of star-crossed lovers is universal, resulting in Shakespeare's Romeo and Juliet, the play and movie West Side Story, and hundreds of other books, plays, movies, etc.

It would be wrong to make a play from of famous book that is still subject to copyright protection. Very old books are no longer protected. Careful needs to be taken with recent translations of foreign language books.

It is absolutely illegal to make a famous book into a play without obtaining the legal rights.

If the book is not in the public domain, it is probably covered under copyright, which would give the owner of the copyright the sole right to make derivative works, which would include a stage play. If the book is in the public domain, such as if it is an older book, then you can make a stage play of it. There are websites that list books that are in the public domain, and there are many excellent ones. Otherwise, if you are set on the one book and it is not in the public domain, get a lawyer to help you and you may be surprised that the copyright owner may let you make a play of it. It is worth a try.

What you are talking about is creating a derivative work, i.e., a work based on the work of another person. If you do this without an assignment or appropriate license you will commit copyright infringement and if the copyright is registered (which is certain to be true for a famous book) the damages are $150,000 per violation as a willful violation. Before doing this I strongly suggest that you communicate with an experienced and qualified IP attorney.