This elevator could be dangerous in survival-mode. I recommend building this on a local world, not on a server (delay times…). Build two towers made of sticky pistons (as high as the elevator/lift should become) (there should be a 4 block distance between the towers).

Add blocks and stairs in turn to the pistons (shifted on both sides!).

Repeat this step for all of the pistons.

Add additional stairs in front of the lowest stair.

Build two towers of blocks, one block behind each piston-tower.

Start adding redstone-torches at the built towers, but be careful: Start adding torches at the side where the lower stairs are (look at image) and add the torch to every second block of the tower. Add redstone-torches, but shifted by one block to the other side (look at the arrows in the image, the redstone torches are at the height of the stairs!).

Delete the blocks between the redstone-torch-blocks (delete blocks without redstone-torches).

Start building this construction, which connects the redstone-torch-blocks.

Add redstone lanes and a repeater (highest delay-level!) to this construction.

Repeat the same construction for the next floor, but mirror inverted!

Repeat these steps until you reach the end of your piston-tower.

…and you already got it, right? Repeat the steps again for the other side…

Connect the floor-signals of both towers.

Add a repeater on the second delay-level (one right-click) to the side, where the first stairs aren’t.
If you finished this step, add a block with a redstone torch on its back. The pistons should move the stairs apart.

Put a button on the other side of the block. Press it to start the lift/elevator.

Job done! Your new elevator should start working now when you press the button.