Alfred Felix Landon Beeston, known for most of his life as Freddie, was born in Barnes in 1911, the only son of Herbert Landon Beeston and Edith Mary Landon. He was elected a King's Scholar at Westminster School in 1925. Whilst there he developed a love of languages and received thorough training in Greek and Latin. Already at the age of fourteen he had become fascinated by South Arabian inscriptions which he had seen in the British Museum. He even tried to transcribe and decipher some of the Sabaean inscriptions there. After gaining his ‘School Certificate’, he was attracted to the idea of a career as a librarian and discovered the existence of the Department of Oriental Books and Manuscripts in the British Library. After finding a second-hand copy of Palmer's Little Arabic Grammar, which he found ‘intelligible and helpful’, he began to learn as much Arabic as he could. When he won prizes at school, he chose a copy of the Qur’an and an Arabic dictionary.

He then decided to study Arabic at Oxford. He received total support and encouragement for this plan from his father. Having taken Prelims in Classics, which had been his special subject at school and the means by which he gained a scholarship to Christ Church in Oxford, he turned to Arabic as his major language and Persian as a necessary minor language. He was delighted to learn Arabic but less keen on Persian. It was his good fortune to take as an extra subject South Arabian epigraphy, which was taught by D. S. Margoliouth, the Laudian Professor of Arabic at that time, ‘whose teaching methods were slightly odd, but suited [him] admirably’. This choice of subject had a great impact on Beeston's subsequent career, as he describes in a typically amusing and informal autobiography that he wrote shortly before his death.

Whilst an undergraduate, he went to talk to the Keeper of the Oriental Department in the British Museum about his desire to work as an Arabist there. After graduating with a first-class degree in Arabic (with Persian) in 1933, he embarked on a D.Phil. He was awarded a James Mew Scholarship in Arabic in 1934. In the summer of 1935, he was appointed as a junior librarian in the Bodleian Library in Oxford, and he also managed to complete his D.Phil. in his beloved field of South Arabian inscriptions in 1937.