He was born the eldest among eleven children of James J. Barry, a tea and wine merchant of Ballyhooly, County Cork, and Annie Barry (née Ryanh). His family had a small grocery business at Bridge Street in Cork.[2] The firm specialised in teas and wines. The business later moved to Princes Street. It was awarded the Empire Cup for Tea Blending at the 1934 Grocers Exhibition in London. He spent his working life at the firm which was later developed under the tutelage of his son Peter into a major company called Barry's Tea.
Barry's son Peter won a seat in his old constituency in 1969, he was a Minister and briefly Tánaiste in the 1980s, and was succeeded as TD in 1997 by his daughter Deirdre Clune. Anthony Barry's daughter Theresa (Terry) Kelly served as Mayor of Limerick from 1983 to 1984.[5]
Barry had an interest in photography from an early age and in his 50s his skill developed and apart from taking photographs he developed them at his home in York Terrace.[2]