He has come to the end of his tether. He has outrun his fortune; he has exhausted all his resources. The reference is to a cable run out to the bitter end (see Bitter End), or to the lines upon lines in whale fishing. If the whale runs out all the lines it gets away and is lost.
Horace calls the end of life “ultima linca rerum, ” the end of the goal, referring to the white chalk mark at the end of a racecourse.