How to soundproof sliding door/wall?
Those look like thin, hard panels and they are much more of the problem than that little gap. While plugging that gap may stop a bit of the sound, the panels themselves are acting as sounding boards and transmitting the sound through.
I have a bookshelf nook in my bedroom wall with nothing but 1 layer of drywall behind the books, and a speaker just the other side of the nook. Sound is actually louder in my bedroom than it is in the other room because that piece of drywall reverberates and moves more air on the bedroom side than the speaker cone does on the other side.
Note that there is very little you can reasonably and somewhat inexpensively do to &quot;sound proof&quot; that wall, especially in something that would be removable when you move out. You can, however, deaden the sound to an extent, and that may make your quarters more livable.
The only realistic &quot;fix&quot; would be to cover that entire wall with loose hanging fabric of some sort to help capture and deaden the sound the walls are transmitting through. Your best bet would likely be to put a curtain rod across the entire set of panels and hang curtains/blankets/strips of cloth across the whole thing. Depending on how much room there is above the door panels, you may have to hang the rod from the ceiling instead of the wall. You may find that it will take a fairly heavy curtain or (set of) blanket(s) to kill off enough sound, so be prepared to buy a sturdy curtain rod and mounting brackets.
Generally speaking, most landlords are OK with a few holes in the wall (or ceiling as may be the case) for hanging things because small screw holes (or drywall/plaster anchors, molleys, toggle bolts, etc) are generally easy to fix up with some Spackle. If you're not sure about doing this, talk to your landlord about it. If he's edgy about the damage, either agree on an amount to be deducted from your damage deposit to cover the repair (get it in writing, just to be safe), or promise to make the repairs to his satisfaction prior to moving out. Or, just be willing to sacrifice your deposit for a bit of peace and quiet.
You could, of course, purchase acoustical panels of the sort used in recording studios and attach them to those sliding panels, and that would work far better than a curtain/blankets, but they'd probably be more expensive, have some sort of (semi-)permanent mounting adhesive to hold them to the panels, and you'd have to have at least some of them on the other side of the wall to allow the door to continue to slide, so that's probably out of the question.