Is it possible for people of the same sex to marry each other?

No. Gay (or homosexual) “marriage” is not really marriage at all. Marriage was instituted by God as a relationship between a man and a woman. “Therefore a man leaves his father and his mother and cleaves to his wife, and they become one flesh” (Gn 2:24). The Church does not have the power or authority to change this—nor does secular society. Even if the legal definition of the word “marriage” were changed to include homosexual couples, those couples would not ever be truly married.