The term originates from a popular anime trope in which characters (almost always female ones) are able to produce ridiculously-sized mallets seemingly from nowhere on command, usually in response to lechery.
Hammerspace Objects and Enchantments
There are several items that produce hammerspace, such as portable holes or Bags of Holding. Because TARDISes are larger on the inside than on the outside, they may be said to have a hammerspace-like quality. Kender are sometimes suspected of having a plothole- or hammerspace-like special ability that attaches to their clothing, due to their habit of pulling things they shouldn't logically possess out of their pockets. Some magic-users are also able to conjure objects from nowhere apparent, which replicates the hammerspace effect. In the Young Wizards-verse, there is a specific spell to create a space-time pocket called a claudication for just this purpose. Power Rangers, members of Super Sentai teams, and Kamen Riders produce their personal weapons and transformation trinkets from hammerspace, and some of the Zords (the individual smaller giant robots that combine to make the larger ones) are stored in hammerspace when not in use. Additionally, most video game characters are accepted to have hammerspace somehow due to the massive (items in their) inventory they store without visible pockets or bags.
Using Hammerspace and Hammerspace as a Power
It is frowned upon for agents to have access hammerspace as a superpower if they are not from a hammerspace-enabling canon (except in circumstances in which it would be really funny), but any agent may hold and use an object that is hammerspace-enabled without any training or background. In fact, Bags of Holding may be more convenient than regular luggage if an agent can get their hands on them. Just ask Agent Trojie.
Characters who are not canonically supposed to have hammerspace occasionally develop it because of an author forgetting to mention the presence of an object before it is used. If the object could reasonably be kept within a pocket or be found in the surrounding area, this is not a problem, but the sudden appearance of a large or unusual object from nowhere for no apparent reason is a charge. For example, even though Murdoc Niccals is a cartoon character, he does not canonically have the ability to produce shotguns from nowhere, a fact which did not escape the MSTers of "Like it or not" (Part 2) (NSFW), in which he inexplicably did so. Nor should characters in the Redwall or Twilight continua be able to pluck their weaponry from thin air, as Zain did with his scythe and Tiana did with her "samurai sword."
Unjustified hammerspace is very common among Mary Sues, mostly because many of them do not think about actually being able to carry around everything they may mention as being on their person.
Mace-space is a phenomenon which allows Agent Kayleigh Leonard (Department of Bad Slash) to produce a two-handed mace at need, despite spending half her time in a bikini. This is similar to the convention that women can produce large mallets at need and may — or may not — be related.