Eskew was one of four sons born to Harry Leroy and Oretha Wilson Eskew. He attended Shades Cahaba School and graduated from Shades Valley High School. He completed a bachelor's degree in civil engineering at Auburn University in 1956 and married his high-school girlfriend, Martha Bonner. That same year he was commissioned as a 2nd Lieutenant in the U.S. Army and was stationed to Fort Belvoir, Virginia. He later served in the U.S. Army Reserve with the 926th Engineer Battalion. He returned to active duty in August 1961 and was assigned to the 101st Airborne Division at Fort Campbell, Kentucky, where he served until his discharge, with the rank of Captain, in 1964.
Eskew and his brothers took over the H. L. Eskew & Sons plumbing, heating and air conditioning business, which was reincorporated as E. B. Construction in 1976. He later worked for Trinity Construction. Eskew served as president of the Birmingham chapter of the Alabama Society of Professional Engineers and the Mechanical Contractors Association of Birmingham.
Eskew was also a noted grower and judge of roses and was an active member and leader of the Birmingham Rose Society. He was recognized as a "Master Rosarian" by the National American Rose Society and served on a number of its committees. He was also recognized as a Certified Master Gardener at the Birmingham Botanical Gardens and was honored as the Gardens' 2006 Volunteer fo the Year.
When Eskew died in 2019 he was survived by his wife, three children, five grandchildren, and one great-grandson. His funeral service was held at Vestavia Hills Baptist Church, and he was buried at Elmwood Cemetery.
- "Robert Leroy Eskew Sr" obituary (April 17, 2019) The Birmingham News