A house; perhaps from the Italian casa. In the canting lingo it meant store or ware house, as well as a dwelling house. Tout that case; mark or observe that house. It is all bob, now let’s dub the gig of the case; now the coast is clear, let us break open the door of the house.
Definition taken from The 1811 Dictionary of the Vulgar Tongue, originally by Francis Grose.
See also the definition in Nathan Bailey's 1736 dictionary of canting and thieving slang.
To Cascade * Case VrowNathan Bailey's 1736 Dictionary of canting and thieving slang
John S. Farmer's collection of canting songs and slang rhymes