Green’s Dictionary of Slang

a small measure or glass of gin, prob. 2d. worth.

1861		(con. 1840s–50s) H. Mayhew London Labour and London Poor I 258/1: I have overheard a proposal for ‘half-a-quarteen and a two-out’ (glass) between a couple of shoeless boys.
1904		Proc. Old Bailey 20 June 702: During the day I had half a quartern of rum and ‘two out’.
1919		Leigh & Collins [perf. Marie Lloyd] Don’t Dilly-dally on the Way 🎵 I was cross about the loss / Same as any other human woman would / But I soon got over that, what with ‘two-out’ and a chat.