Green’s Dictionary of Slang

starry n.

In phrases

do a starry (v.) [SE star]

(UK Und.) to sleep in the open air.

1926		N. Lucas London and its Criminals 226: Ullo, Muvver, what’s up, doin’ a starry?
1933		M. Marshall Tramp-Royal on the Toby 178: To be a-doing of a starry in London after midnight is hardly a thing to be made jest of.