You can think of the cubed items as extra 'passives' of affixes that you can activate and deactivate at will, for no cost at all, as long as you had that item and 'cubed' it (which would destroy the item). And it is 'character-type' bound. It's across your account but different on season vs. non-season (and hardcore vs. softcore on each), just like artisans level and their recipes, paragon levels, blood shard caps, etc.
So if you have a Furnace in the cube on your non-season softcore Wizard, your non-season softcore Barbarian can activate it, too. No need for this Barbarian to cube another Furnace. Your season characters, on the other hand, have to find their own Furnace to get into the cube.
The only restriction is that you can only activate affixes from items that the character can equip. So in the example above, your non-season softcore Crusader can activate the Furnace's affix, but your non-season softcore Demon Hunter can't, because DH cannot equip the Furnace.