connection, barrier, gateway, involve, concerning
關 centers on a point of controlled passage: a gateway or barrier that connects two sides. From this, it extends naturally to involvement and concern, as things become linked or implicated through such a passage.
關 combines 門 (gate) with a phonetic component that also suggests a bar or bolt. The character originally depicted a gate with a crossbar, representing a barrier or a point of connection that can be opened or closed.
The outer 門 is a gate, and the inner part looks like a bar or bolt. Picture a heavy gate with a crossbar that controls passage: it is a barrier, but also a connection point where things become involved.
For カン, imagine a can of oil used to lubricate the rusty gate so it can open and close smoothly: can -> カン, and the gate is the barrier that connects two sides.