I'm presuming it's brick, Dave? At the top of the thread is a photo of a garden shelter near me, where the back wall was reconfigured into a window postwar. Think it's only used as a shed, but would make a great man-cave! Various local authorities provided brick shelters to tenements (one between two families), and middle-class families often built their own. I reckon if you waterproofed the structure's fabric appropriately, and let the inside dry out, it would whitewash quite nicely.