You can do a few things if you decide to turn that closet into a dedicate gear space...
The doors you have are going to do nothing. If you remove them, I would wager the sound is almost the same level... just a slightly different pitch. If you can replace those doors with solid doors on hinges, with a seal all the way around, you'll eliminate a LOT more sound.
You can also put another layer of sheetrock inside the walls adhered with green glue, that'd help also.
If you do both of those and add some insulation bats in fabric to the backs of the closet doors, you'll get it pretty much silent. You'll need to ventilate it to an outside wall or the ceiling if you do this also...
Barring all of that, I would build a box out of MDF and then layer that with green glue and then rap another layer of MDF around that. Put a door on the front with the same MDF/GG/MDF combo and you'll have a pretty silent box to put things in... (also known as a 'hush box')
Just some thoughts. Basically, your closet has no sound insulating properties now and lots of air (sound) leaks. Folding doors won't prevent sound from coming through as you've found... So, that's the area I'd attack first...
-C