Meyer was the son of a wealthy New York family. His father, Cord Meyer Sr., was a diplomat and real estate developer; his mother, Katherine Blair Thaw, belonged to a Pennsylvania family that earned its wealth in the coal business. His grandfather, also called Cord Meyer, was a property developer and a chairman of the New York State Democratic Committee. He was educated at St. Paul's School, New Hampshire, and attended Yale University, where he was a member of the Scroll and Key society. After graduating in 1942, he joined the 22nd Marine Regiment and fought in Pacific War; he took part in the Battle of Eniwetok, and in the Battle of Guam as platoon leader, losing his left eye in a grenade attack. He became a first lieutenant and was awarded the Bronze Star Medal. He shared his war experiences, writing for "The Atlantic Monthly".