I found that one part of the feeling you describe comes from it being "organic" in an unusual way. It's easy to find any number of modern buildings that use "organic" shapes or materials (think curved or made from wood). However, they will still be boring and predictable inside and fit into a very boring street grid outside. Barbican is different: it's "brutalist" and as un-organic in its material as possible, but the structure itself is an unpredictable, multilevel maze where you don't know what awaits you around the corner. That provokes a very different feeling, more akin to a forest over the rolling hills than to a city. And it's built that way both inside and out, at least as far as public spaces are concerned.

I think that different kind of "organic" is why I love Barbican. There aren't many places that make me feel like a child exploring a new area, but Barbican is the best at that.