Born in Manchester his family shortly afterwards moved to Liverpool. He obtained a B.Sc. in Physics from the University of Liverpool in 1929 and a D.Sc. in 1933. After graduating he worked in what was then the new subject of X-ray diffraction, together with Henry Lipson.

The two scientists frequently asked for advice from Lawrence Bragg, who at the time was working at the University of Manchester. He eventually went to work at Manchester and Hull before becoming Dewar Fellow in Crystallography at the University of Edinburgh in 1938, he was elected Fellow of the Royal Society of Edinburgh in the same year.

Shortly after arriving in Edinburgh, the Second World War started. Being a member of the Society of Friends and a conscientious objector, he was sent to work with Professor Norman Dott at the Western General Hospital to help work on electroencephalography. After the war ended he joined staff at the Department of Chemistry and built up an X-ray diffraction laboratory there.

As well as scientific research Arnold Beevers was also a teacher as well as being involved with the lives of disabled people. His last contribution to science was the creation of Beevers Miniature Models. He was never happy with digital computers and he worked to create accurate ball-and-spoke models, to a smaller scale than previously available. He worked out a way of creating models which he thought must be both accurate and elegant. From the beginning of his work he had relied on work carried out by disabled workers to create the models, they took great pride in the work they carried out for him.

He and his colleague and friend, Henry Lipson, are best known for the Beevers-Lipson Strips a technique by which they reduced huge calculations of three-dimensional Fourier summations to sums of a more manageable one-dimensional one.