From The Phrase Finder:
“*Euphemism for being bankrupt or in debt.
This phrase originates from the London street where the UK bankruptcy court used to be located. The court moved to Carey Street in the 1840s but the phrase didn’t emerge as a synonym for bankrupt until much later. The first reference I can find is a piece by James Agate, in The Saturday Review from 1922:
“The melancholy gentleman in direful Carey Street.”
It is sometimes suggested that the term Queer Street is a slang synonym for Carey Street. That’s not so however, as the use of that phrase pre-dates the court’s move to Carey Street and, in any case, the two phrases don’t mean quite the same thing.
Pascal Treguer writes at word histories:
“‘QUEER STREET’, WHERE PEOPLE IN DIFFICULTIES RESIDE
The name Queer Street, or Queer street, queer street, originated in British-English slang in the early 19th century; it denotes an imaginary street where people in difficulties, now especially financial ones, are supposed to reside—it is, so to speak, the urban counterpart of Dicky’s meadow, a Lancashire term also dating from the early 19th century.
It is first recorded in Lexicon Balatronicum. A Dictionary of Buckish Slang, University Wit, and Pick Pocket Eloquence (1st edition – London, 1811), an expansion by Hewson Clarke (1787-circa 1832) of A Classical Dictionary of the Vulgar Tongue, by Francis Grose (1731-91):
Queer Street. Wrong. Improper, Contrary to one’s wish. It is queer street, a cant phrase, to signify that it is wrong or different to our wish.
The earliest actual use of Queer Street that I have found is from The Globe (London) of 2nd December 1825, which gave an account of a boxing match between ‘Young Gas’ and Morris Pope, which took place in “a field near Devil’s Ditch”, near Andover:
Round 3. Gas let fly right and left, giving Pope a tremendous blow over his loft ogle, putting him a little into Queer street, and leaving a small incision, about an inch and a half long, to tell the “flattering tale;” Gas napt it on the ribs at the same time. Pope down.
The second-earliest instance that I have found is from a paragraph in The Sydney Herald (Sydney, New South Wales, Australia) of 5th December 1831:
Catherine Anderson, picked up, as the constable described it, in “queer street,” was sent 14 days to the factory*.
* Here, the factory designates Parramatta, in New South Wales, a detention centre for women, where cloth was manufactured by inmates.
It has been said that Queer Street is an alteration of Carey Street, the name of a street in London, site of the bankruptcy court. But this derivation is impossible, since the court was not located there until after 1840; besides, Queer Street did not appear in the restricted sense of financial difficulties.”