Short answer: less than 50 years old at the time of this post. You may have heard that the Enneagram personality types are an ancient system going back thousands of years. I heard that too, but I looked for proof for the past 20 years or so. I found nothing except proof to the contrary.
Some of those authors that I first heard the ancient system story from have since changed their tune and offer something more verifiable. Others continue to use it. It seemed to be one of those things that people repeated from teacher to student over time and it just hasn't completely gone away yet. What convinced me beyond any doubt to be done with the story was hearing Claudio Naranjo describe in a YouTube video that it was made up to simply give the types more credence. He uses a quote by Oscar Wilde to exemplify why the ancient system story was made up "If you want an idea of yours to become famous attribute it to a famous person." You can watch the video for yourself.
Another reason why people may still think of the Enneagram personality types as ancient is that people tend to call the system "the Enneagram." The Enneagram is actually just a symbol which has nothing to do with the types. That symbol was first introduced to the western world by Gurdjieff in the early 20th century. He purportedly learned of the Enneagram symbol from a Sufi brotherhood. To speculate that the symbol has ancient origins is quite another matter from speculating that the personality types have ancient origins. Gurdjieff didn't use the symbol with personality types. Naranjo was quite clear about that in another video.
What happened was Oscar Ichazo began using the Enneagram symbol with ego fixations sometime in the 1950s and 1960s. He didn't use the symbol in the Gurdjieff tradition. He started a new use of the symbol (I describe it in detail in the Enneagram User Guide). The history of the symbol as traced back to Gurdjieff and before has no relevance to the history of the Enneagram personality types. They're two separate things.
So, here's the factual history of the Enneagram personality types. Ichazo placed ego fixations and such upon the Enneagram symbol during the 1950s and 1960s. Where he found the symbol and the material he incorporated into his work is unclear. He studied a wide number of traditions that he could have drawn from. In 1970, Naranjo learned of Ichazo's enneagram material. Ichazo called that material proto-analysis but Naranjo couldn't use that term because it was copyrighted or trademarked. Naranjo began using Ichazo's proto-analysis material as part of some larger work he was doing with people and began referring to the types as enneatypes. Other people learned about the enneatypes and began using them. They became popularly known as the Enneagram personality types which became shortened to the Enneagram.
The Enneagram personality types are as old as Naranjo's enneatypes which he derived from the proto-analysis material he learned from Ichazo in 1970. Is the system ancient? No. Do elements of the system have ancient origins? Perhaps. Both Ichazo and Naranjo naturally drew from their understanding of all the traditions they had studied.