Educated at Bangor Grammar School and Queen's University where he studied structural engineering with architecture, Stuart McCloskey played club rugby for Dungannon RFC and joined the full Ulster Academy ahead of the 2013-14 season.

He made four senior Ulster appearances that season and was named Academy Player of the Year at the 2014 Ulster Rugby Awards, playing for Emerging Ireland and establishing himself as a regular starter for Ulster before winning his first senior international cap for Ireland against England at Twickenham in 2016 and receiving the Ulster Player of the Year Award.

While he found it difficult to break into the Irish Six Nations squad following his debut, his persistence and continued sterling performances for Ulster, for whom he has now made 217 appearances, paid off, and he was one of only two Ireland players who played in every minute of all five games in the 2026 Six Nations Championship.

His display against England at Twickenham prompted Irish coach Andy Farrell to say "He should definitely be in the running for player of the tournament!" Farrell hailed his try-saving cover tackle in the win against Marcus Smith of England as a defining moment in the Championship.