That's easy once you internalize the effects of gravity on the Euclidean curvature of space! The relationship between a volume and its surface area is set by the local curvature, not a fixed constant. π is just the value for a flat 2-dimensional space. That means that if you have a 'neck' with negative curvature, followed by a correction cusp with positive curvature, you can have a locally-flat region with a larger internal volume than the surface area of the opening of the neck would allow in a flat spacetime.
This is difficult to model intuitively, although of course the math works out, unless one thinks about space as being embedded in a higher-dimensional manifold, allowing the space to 'bulge' out. Because of relativity, those two perspectives (warping space itself vs. allowing it to move inside a larger manifold) are actually interchangeable, but the second visualization makes it much easier to understand how this still falls under the same concept of 'motion': in order to make a space that is larger on the inside, you have to allow objects to move slightly in a direction perpendicular to all four of our native spacetime, although of course this motion is not actually observable from inside the embedded space. Instead, you just get slightly stronger time dilation across the neck of the volume.
That's actually also how wormholes work, except that they connect back to the same locally-flat space, instead of having a cusp and a separate internal space.
The hard part, if you think about it, is not so much creating an object that's larger on the inside, but rather ensuring that the objects you put in and take out don't explode into an energetic cascade of exciting particles as their internal geometry is warped by passing through the neck. Fortunately, objects are mostly empty space, so if you flatten the neck along one axis enough, you can have a minimal 'plane' of distortion that doesn't introduce enough extra energy to the electromagnetic field to cause large disruptions to the objects being stored.
... of course, if one's control is insufficiently precise, that scheme will produce a storage space where traversing the neck tears apart and disrupts very small-scale structures, such as the ones necessary for living things to continue living. An energy-based analogue of Nyquist's sampling theorem means that you need to get the thickness of the neck down to less than half of the smallest distance between two constituent particles that you want to remain bound to each other. So, for practical purposes, your neck can't really be much thicker than one angstrom.
And no version of this scheme can make it safe to pass femptotech or other gravitational distortions through the neck. So if you tried to pass one neck through another neck you'd get a messy explosion.