1.
barrier; checkpoint; gate
A physical checkpoint or gate that controls passage, such as a border crossing or toll gate.
2.
hurdle; obstacle; critical test
A figurative barrier that must be overcome to proceed, such as an exam, interview, or difficult stage in a process.
Usage
The figurative sense (hurdle, obstacle) is more common in everyday conversation. 関門 implies a barrier that must be cleared before proceeding — it suggests a test or trial rather than just any difficulty.
関門海峡 (Kanmon Strait) between Honshu and Kyushu is a well-known proper noun using this word.