Consider a visual experience, E, as of a yellow flash. Associated with E in the cortex is a complex of neural structures and events, N, which does admit of spatial description. N occurs, say, an inch from the back of the head; it extends over some specific area of the cortex; it has some kind of configuration or contour; it is composed of spatial parts that aggregate into a structured whole; it exists in three spatial dimensions; it excludes other neural complexes from its spatial location. N is a regular denizen of space, as much as any other physical entity. But E seems not to have any of these spatial characteristics: it is not located at any specific place; it takes up no particular volume of space; it has no shape; it is not made up of spatially distributed parts; it has no spatial dimensionality; it is not solid. Even to ask for its spatial properties is to commit some sort of category mistake, analogous to asking for the spatial properties of numbers. E seems not to be the kind of thing that falls under spatial predicates. It falls under temporal predicates and it can obviously be described in other ways - by specifying its owner, its intentional content, its phenomenal character - but it resists being cast as a regular inhabitant of the space we see around us and within which the material world has its existence. Spatial occupancy is not (at least on the face of it) the mind's preferred mode of being.
No doubt this is connected with the fact that conscious states are not perceived. We perceive, by our various sense organs, a variety of material objects laid out in space, taking up certain volumes and separated by certain distances. We thus conceive of these perceptual objects as spatial entities; perception informs us directly of their spatiality. But conscious subjects and their mental states are not in this way perceptual objects. We do not see or hear or smell or touch them, and a fortiori do not perceive them as spatially individuated.(2) This holds both for the first- and third-person perspectives. Since we do not observe our own states of consciousness, nor those of others, we do not apprehend these states as spatial. So our modes of cognition of mental states do not bring them under the kinds of spatial concepts appropriate to perceptual acquaintance. Perceptual geometry gets no purchase on them. And this is not just a contingent fact about the mind.(3)
Nor do we think of conscious states as occupying an unperceived space, as we think of the unobservable entities of physics. We have no conception of what it would even be to perceive them as spatial entities. God may see the elementary particles as arrayed in space, but even He does not perceive our conscious states as spatially defined - no more than He sees numbers as spatially defined. It is not that experiences have location, shape and dimensionality for eyes that are sharper than ours. Since they are non-spatial they are in principle unperceivable.
This is I think what people have in mind when they aver that 'consciousness is not a thing'. The thought expressed here is not the trivial one that to refer to consciousness is to invoke a category of events or states or processes and not a category of objects or continuant particulars. Our intuition that conscious states are not spatial is not the intuition that no state is an object. For ordinary physical states and events are spatial entities in the intended sense: we apprehend events as occurring in space, and states are features of spatially constituted objects. So it would be wrong to offer a deflationary interpretation of our non-spatial conception of consciousness by insisting that it comes to nothing more than a recognition that talk of consciousness is talk of events and states - just like talk of explosions and motions and electric charge. The non-spatial nature of consciousness, as we conceive it, is much more radical than that diagnosis suggests. Descartes was not committing the simple howler of failing to notice that conscious phenomena are not objects at all and hence not spatial objects. In fact, even when we do speak of something that belongs to the category of continuant object, namely the subject of consciousness, we are still insistent upon its non-spatial character.(4) The self is not a 'thing' either, in the intended sense. The realm of the mental is just not bound up in the world of objects in space in the way that ordinary physical events are so bound up. So, at any rate, our pretheoretical view assures us.