As Eddie advises the game loads all players stash into memory thus 350 x 4 is a lot of items.
They did advise they would increase it but retracted it because it caused lag and load times to be too long (this is why the game takes a while to load).
Thus the only opton is to create another character, go into your stash and pick up 60 items thus each character can hold 60 items.
Bit of a pain if you need to navigate through it but nothing can be done.
D4 is the answer with online saves.