Baroka is the eponymous lion of the play's title. He is known by this name because of his strength, but as we meet him in the play, he is over sixty years of age, and his strength is diminishing. Baroka is the chief of a village, and he has many wives. He is not used to women laughing at him, and neither is he used to women turning down his offers of marriage.
Baroka notices a beautiful young woman named Sidi, and he sends his most senior wife, Sadiku, to propose an offer of marriage to Sidi. Sidi laughs and turns down the offer because she thinks that Baroka is much too old for her. When Sadiku tells Baroka that Sidi has turned down his offer because of his age, he becomes upset, and he begins to feel sorry for himself. It is in this mood that he tells his wife that his "veins of life run dry" and that his "manhood is finished."
When Baroka says that his "veins of life run dry," he means that his blood is no longer young. When he says that his "manhood is finished," he means that he is no longer as strong and as masculine as he used to be. The two images together also perhaps imply sterility and impotence. Baroka is no longer a virile, exuberant young man, and Sidi's rejection makes him realize this.