In Co/CoO nanostructures, of dimensions l×3l, at small Co thickness (≈6,10 nm), a strong increase in the bias field and the associated coercive field are found as the nanostructure size is reduced from l=120 nm to l=30 nm. This property indicates that the characteristic length D(AF) within the antiferromagnet which governs exchange-bias effects is the nanostructure size. By contrast, at larger Co thickness (≈23 nm), the exchange-bias field does not depend on the nanostructure size, implying that D(AF) is smaller than the nanostructure size. The results are discussed in the framework of the Malozemoff model, taking into account that the coupling between CoO grains is weak. Exchange bias is dominated either by coupling within the antiferromagnetic layer (6- and 10-nm-thick Co samples) or by ferromagnetic-antiferromagnetic interfacial coupling (23-nm-thick Co sample).