In any case you choose it will have to pass the safety standards of a goods elevator. This includes being power fail safe, actuator fail safe and to have positive braking. Thus you are either going to have a regular cable system or chain with a motor, brake and gearbox driving a wheel, along with a counterweight, or a hydraulic system with a pump, fluid reservoir and a set of lock valves to hold the car in position.
As well you will need guides, and whatever you choose the standard safety design consideration is that it must not fail at a 12 times overload, and must still operate with a 2 times normal max load in it.
Sensing as well must have at a minimum a sensor per level, along with a mechanical overrun switch placed at close to max and min travel, along with an emergency power disconnect ( which shorts the power rails to trip a breaker or blow the main fuses in the supply) just before hitting the main bump stops or emergency dampers.
Your controller must have documented code, and must not fail if any inputs are wrong, but must detect external input faults and fail safe to not operating the motor. It must also have protection from faulty code, which typically is a watchdog timer that when it times out resets the system and locks the motor drivers into a non operating state. Controller inputs and outputs must meet regular industrial input and output protection ratings, and must survive connection to mains rails for long periods without failing. This typically means opto isolated inputs and relay outputs. Power supplies as well must be designed for long period operation with the ability to survive all the transients that come in on the mains, including massive overvoltage events and lightning strikes. thus typically you find still a mains transformer feeding a bridge rectifier, along with a self resetting thermal overload in the transformer. Motors must have current limiting and overheat protection to protect them. Motor breakers and physical latches must have a safety link that opens on failure so that any fault there results in no movement. this circuit must be a physical break of the motor's drive signal, not a software only operation.
You need door detection, along with detection of objects in the openings, and switches to show both door open, door closed and door locked status, along with having all door sections having a switch, both on the platform and on the outside. Whatever motor drives the door must have protection as well.