To perform this you need a throwable item - you can pick up one to the left of the lever. Next - you need to get to the lower level to the machines, where girl and guard are located, as quiet as you can - because if you start panic too early you will mess up enemy movements and they might gang up on you in door frames and tight corridors.
When you are close to the machine - you can either bump into girl OR you can Sprint next to her - she will get scared and alert the guard, moving him away from door frame, which is really helpful.
Then you do the Storage trick, and this time we store the closet's opening animation. You need to carry this storage to the VIP room on 2nd floor. While sprinting - pay attention to the surroundings and keep track of girls and guards - some of them might bump into you in panic or to attack you. If you need to recover stamina - let go of all buttons except Sprint - that button allows you to keep the storage while recovering stamina or when running.
When you are in the VIP room, look for this set of pillows:
It allows you to climb up a bit to a place where you will perform Storage, twice. Align yourself like this:
It's not very precise, but you don't want to be too far off from that arc in front of you.
And now the most important part - you have to:
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press Action button - to activate Storage for the first time.
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DO NOT barge out of it - slowly exit it out of it.
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Don't press any buttons while you exiting it - that will activate the 2nd Storage.
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And again - DO NOT barge out - slowly get out of it - it will force you OoB.
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WHEN YOU ARE FALLING DOWN YOU MUST HOLD "S" (walk backwards) BUTTON - not doing it puts you in risk of falling down into abyss.
Once you are oob and alive - run to the corner (1:07 on the video) and align yourself perpendicular to the wall and start slowly walk forward - as soon as you feel like you fell down a bit - wall backwards for 1s (the reason why you "fall" there - you land on the top of the pillar, and it's lower level then you)
Once you walked backwards enough - start Sprinting straight forward until you land - once you land - take this path (red):
With blue i have marked the area where you will get stuck completely until you restart checkpoint. Avoid it. It just takes practice. You don't have to sprint here, but it is possible to do rest of this OoB while sprinting blindly after you landed from previous jump.
After you land down you need to hit the checkpoint that is located above you - as soon as you seen autosave symbol pop-up - press F9.
P.S. If you need throwable for next Storages - watch rest of the video for the nearby location.