Babaji is an unascended Master of the Himalayas, who became well known in the West through Paramahansa Yogananda's book "Autobiography of a Yogi".
Babaji chose not to take his ascension/final liberation by reason of the Bodhisattva vow, which means he desires to stay on earth and assist mankind until everyone has gained his/her immortal freedom.
Yogananda says, "Babaji is ageless. No one knows how old he is."
Some people say he is hundreds or thousands of years old… His body shows no sign of ageing and he needs no physical food - therefore he seldom eats. Also, he is only recognized if he wants to be.
My personal experiences with Babaji came as a result of practicing the Kriya Yoga
meditation and breathing techniques brought to the West by Paramahansa Yogananda
and taught as written lessons for beginners and also under the guidance of qualified swamis of the Self Realisation Fellowship.
For more info I recommend the books "Autobiography of a Yogi" by Paramahansa Yogananda and "Babaji: The Divine Himalayan Yogi" by Swami Satyeswarananda Giri