Up a blind alley
Following a course of action that leads to no good outcome.
Alleys are blind if they have no 'eye' or through passage - what might now be called a cul-de-sac.
The phrase was first recorded in Richard Stanyhurst's, Thee first foure bookes of Virgil his Æneis translated, 1583:
"Through crosse blynd allye we iumble."
The figurative use of the term, i.e. one where no actual alley was being referred to, came into use in the mid 19th century; for example, this piece of purple prose from the Burlington Hawk-Eye, October 1874:
"If, on account of these things, the overthrow of Republicanism must come, as come it will, we had better reach our destination as quickly as possible, making the issue sharp enough to be understood by all the world, and not be piddling along from year to year in a miserable blind-alley of partisan passion and falsehood, getting weaker and weaker, and poorer and poorer, and madder and madder, under incessant proscription ana vilification."