Stach was born in the town of
Buchholz in der Nordheide, Germany, in the
Hamburg Metropolitan Region and joined Buchholzer FC during his childhood. Since 2010, he participated in training at
Werder Bremen once a week, before he joined the youth academy of Bremen-based Nachwuchsleistungszentrum one year later. Stach played for Werder Bremen until the B-youth, and became captain of Werder Bremen's under-16 side. He later played at JFV Nordwest. Afterwards, he joined the U19-team of
VfL Osnabrück. For the 2017–18 season, Stach joined
SSV Jeddeloh from
Regionalliga Nord, a regional league at the fourth-tier of German football. Stach helped the team achieve promotion to the
Bundesliga. In July 2021, Stach joined fellow German club
Mainz 05 on a deal until July 2024. Later that year, on 4 December, he scored his first Bundesliga goal in a 3–0 victory over
VfL Wolfsburg. On 1 September 2023, Stach signed a four-year contract with
TSG Hoffenheim. Two months later, on 4 November, he scored his first goal for Hoffenheim in a 3–2 home defeat against
Bayer Leverkusen. He finished the
2023–24 season with 408 duels won, the second most only behind
Bernardo. On 22 July 2025, Stach moved to England, joining
Premier League club
Leeds United on a four-year contract worth reportedly £17 million. ==International career==