There are two ways to think about this. First, in the easy way, where you consider matter separate from the space it occupies, you can say that, yes, your meter stick expands -- or rather, starts to, but every time the space between the atoms and molecules increases by a hair, the strong forces between the atoms and molecules immediately pull them back to the correct equilibium distance, and nothing changes overall. You might say the meter stick is continually sliding back into equilibrium as the underlying space expands.
The other way, more true to Einstein, looks upon matter as just a manifestation of the local curvature of space. From this point of view, the expansion of the universe is just the universe becoming, on average, less dense. But as it turns out, matter resists becoming less dense -- because it attracts itself. So the way the universe on average can become less dense is if the reduction in density is confined to those regions that are already nearly empty. Regions in which matter is already dense stay dense. The inhomgeneity of the universe steadily increases. Indeed, from this point of view the universe ends up as an infinitely dilute gas of black holes, with every bit of the universe either perfectly empty (between the black holes) or infinitely dense (within the black holes).