The V&A East Storehouse in East London, situated in a long warehouse building that served as the London 2012 Olympics Media Centre, is a beguiling beast. With its soaring 20-metre-high central Weston Collections Hall and its dazzling storeys of racked storage bookended by wide sociable walkways with see through glass balustrades, it is a 21^st-century temple to artifacts. “Metaphorically, the idea was about fitting as much stuff into that package as possible and then coring out the middle, almost like a geological material, so you can see a cross-section of everything,” explains Elizabeth Diller, co-founder of Diller Scofidio + Renfro, the New York–based architecture firm behind the new museum’s design.

The architects and museum...