Yes, an Xbox Live Gold subscription was required to access almost all internet features on the Xbox 360, including the web browser (Internet Explorer app), online multiplayer, streaming apps (Netflix, YouTube, etc.), and marketplace downloads.
Note: Xbox Live Gold has been succeeded by Xbox Game Pass Core on modern consoles. The Xbox 360 marketplace and online services are largely deprecated, though some peer-to-peer multiplayer still functions for backward-compatible titles played on Xbox One/Series X|S.