One way is to connect an RS flip-flop to each door switch. This is the basic memory circuit. Uses two transistors and a few resistors.
It has two inputs. A pulse at input #1 sets it. After the pulse is ended it makes no difference, the circuit remains in that state. If you apply further pulses at input #1, it remains in that state.
To make it change state, you must apply a pulse at input #2.
The states are symmetrical. Set and Reset are mere nomenclature.
To get a signal to feed to the rest of your circuit, you would tap at one of the transistors.