Plain English Meaning
The airport is shut down and nobody knows when (or if) it will reopen. Until that changes, you cannot plan to fly to or from it, and the published procedures for using it may no longer be valid.
Full Definition
A formal status assigned to an airport that has been closed for an undetermined length of time, with no scheduled reopening date. When an airport is in indefinite closure status, it is removed from active use for normal operations, its instrument approach procedures and departure procedures may be suspended or withdrawn from publication, and the airport is flagged in aeronautical information sources so pilots do not plan to use it.
Context Anchor
Seen when reviewing airport information or instrument procedure material for an airport that has been marked closed indefinitely.
Derivation
"Indefinite" comes from Latin indefinitus, meaning "not bounded" or "without a fixed limit." Here it points to the open-ended nature of the closure -- there is no defined end date, not just a long one.