6v battery low voltage cut-off circuit
The usual way of doing this is to use a comparator circuit that compares a reference voltage with the battery supply voltage. If the battery supply voltage falls below the reference voltage, the comparator disconnects the load e.g. with a P-channel FET. 

As for the reference, you can use a simple zener diode (which is not a good reference) or better a voltage reference (e.g. TL431). Because the comparator also needs to be powered with the battery (down to 5V), you need to use a much lower reference than 5V. So you could e.g. take a 2.5V reference and compare that with the comparator against a voltage divider with 1:1 resistor ratio from the battery.