No, you could use a conductive rail, like a subway, and rack and pinion system to move the elevator. The rack and rail would add a fair bit more total weight to the building compared to a cable. But more importantly, the motors would have to be much much more powerful! Modern elevator systems have a counter-weight balanced on the other side of that cable, which means the motor only has to overcome friction and the small difference in weight between the elevator and counterweight (which varies depending on current payload). The motor on an elevator like Noah is suggesting would have to provide enough force to counteract the entire weight of the elevator + payload + motor + friction, which is at least an order of magnitude more than a traditional elevator.
Let's not forget that rack and pinion elevator cars are significantly noisier, slower, and have much more vibration compared to traditional cable or hydraulic elevators. Rack and pinion is great for portable elevators but a poor choice for a short building, and an awful choice for a tall building.