Four-Dimensional Bags are a series of Magic Items that take the form of small bags.
The following contains plot details/spoilers, read at your own discretion:
Magical Girl Raising Project
After La Pucelle's death at the hands of Cranberry during Unmarked (Arc 1), Fav send a message to the Land of Magic. He blamed the death on an accident and demanded that Cranberry be given magic items under the pretext so that she could better protect the remaining participants.
The Land of Magic send five items, including the bag. Fav and Cranberry would, however, not use them to protect the other participants. Instead they wanted to use them to make the fighting even bloodier. Fav set up an online shop where participants could buy the items in return for their life span.
Calamity Mary bought the Four-Dimensional Bag to help her carry around her rather large arsenal of weapons, including guns, riffles, pistols, grandees, land-mines, knives and harmful substances. She took advantage of having almost immediate access to all of her weapons during her fight with Hardgore Alice and later during her fight with Ripple and Top Speed.
After Calamity Mary's death, Ripple took possession of the bag. She would carry it during her battle against Swim Swim and use one of the weapons stored in it, a flash-grenade, to win the fight.
After Magical Girl Raising Project
Magical Girl Raising Project: Limited
Magical Girl Raising Project: JOKERS
Grim Heart used her bag to transport her army of Shufflins. She would also use it to capture Magical Girls so Joker Shufflin could kill to to replenish her magic when necessary. Kafuria, Umbrain, Princess Quake and Princess Tempest were captured and killed this way.
Snow White would eventually use a similar strategy against Grim Heart. While leaving the facility, she put an injured Filuru inside her bag. When Grim Heart attempted to close the facility door, Snow White told Princess Deluge to also get in the bag. She then threw the bag through a small opening that she made in the door, bringing both Deluge and Filuru to safety.
Some time after the events of Jokers, Snow White regained possession of the bag.
Magical Girl Raising Project: ACES
Snow White once again executed the strategy of carrying a person inside the Four-Dimensional Bag twice in ACES.
First time, she put Premium Sachiko inside the bag while she, Uluru and Sorami Nakano were escaping from Shufflin Ⅱ, CQ Angel Hammer, Dark Cutie, Glassianne and Monoshiri Mi-Chan. In the end, they managed to escape and Sachiko remained unharmed because of the strategy.
Second time, she once again put Sachiko in the bag, this time while traveling to back to Puk Puck's mansion. When they were almost there, Ripple suddenly appeared. Overjoyed at seeing Ripple, whom had been missing for a while, alive, Snow White embraced her. However, Ripple, whom was under Pythie's brainwashing, then took the opportunity to stab into the bag killing Sachiko.
It can store countless objects that you're able to carry in a bag. It is also able to carry living objects, like Magical Girls.