I must agree that the idea of hangar-sized pockets being eventually realized through Star Trek technology is pretty cute. Something for the 25th century, perhaps?
Any physical armor in Trek would have to feature almost comparable levels of magic, really, as there are some pretty severe laws of physics to deal with. A high velocity bullet impacting on armor will pulp the wearer regardless of the strength of the armor, unless the kinetic energy (or the momentum, really) is somehow dissipated. And such dissipating would be extremely difficult, especially if the armor is lightweight and thus has very little inertia.
Thankfully, Trek spacecraft propulsion already heavily relies on the ability to negate inertia and render the law of conservation of momentum irrelevant. But a vest of physical armor, or a bullet-stopping riot shield such as in ST5, would have to be hooked up to an inertia-damping doodad, or it would offer no protection at all even if it were utterly impenetrable. I wonder where those doodads were in the ST5 case...
The ability to sew an inertia-negating gadget into an uniform would seem to negate the need for rocket boots and make it unnecessary for our heroes to stay attached to the ground in any given situation. So we probably have to believe that such gadgetry in 23rd and even 24th century is of somewhat low performance, and better performance comes only from excessive bulk.