An option might be to go Matrix style and extrapolate on the current trends:
Mankind falls in love with virtual reality and technology. By the 2030s, every human was equipped with immersive enhanced reality rigs. Landscape and people are photoshopped in real time, rigs generate a wide range of sensations for their users.
The virtual side of "reality" becomes more attractive while the "real" side become stalled and inhospitable. Pollution and environment degradation push people's lives ever faster towards full virtual reality environment. Life in the "real world" becomes increasingly irrelevant. Work happens exclusively in VR, interaction with the physical world being the purview of specially built robots controlled from a fully distributed VR environment and abstracted away.
People's "real" bodies end up docked in 24/7 in "safe" facilities while they go about their days in a completely virtual environment. Within a couple generations, everyone has forgotten entirely about the existence of the physical universe and of their own bodies. Machines handle the reproduction duties and plug new human units on a regular basis to replace existing worn-out human units.
By the time your story begins, the "real world" is a virtual world and no longer subject to the same laws of physics.
For all we know, it has already happened. ;)
Now, you can have your pockets full of monsters and if you want to store them in some quaint old-fashioned, grapefruit-sized, red-and-white container for the sake nostalgia, who is to tell you different?
Our current reality and its laws of physics didn't account for this perfectly reasonable application, but Reality® Pack Version 3.5b2.08 makes it a breeze.
Think like a manager. If reality isn't consistent with your requirements, reality must be wrong. You just have to upgrade reality.
The scary part is that it's a whole lot more realistic solution than storing genetic information and 3D printing monsters on demand or extracting monsters from a pocket dimension of some kind.