Question about storing magnetic parts (DC motors and speakers)
Modern magnets won't suffer from this problem and older magnets only suffer from this when the threat magnet is producing an opposing field to the victim magnet so, if you feel a degree of repulsion between threat and victim, then it's probably a good idea to segregate them especially if they are old magnet materials.
On the other hand, if you segregate the magnets with mild steel plates, you are unlikely to get a problem.

I wanted to know if there's any problem with just sticking them all
together (they'll literally stick together because they're magnetic)
and putting them in a box.

If they stick together then you are not imposing opposite fields and there will be no demagnetization in this respect.