Disordered hyperuniform heterogeneous materials are new, exotic amorphous states of matter that behave like crystals in the manner in which they suppress volume-fraction fluctuations at large length scales, and yet are statistically isotropic with no Bragg peaks. It has recently been shown that disordered hyperuniform dielectric two-dimensional (2D) cellular network solids possess complete photonic band gaps comparable in size to photonic crystals, while at the same time maintaining statistical isotropy, enabling waveguide geometries not possible with photonic crystals. Motivated by these developments, we explore other functionalities of various 2D ordered and disordered hyperuniform cellular networks, including their effective thermal or electrical conductivities and elastic moduli. We establish the multifunctionality of a class of such low-density networks by demonstrating that they maximize or virtually maximize the effective conductivities and elastic moduli. This is accomplished using the machinery of homogenization theory, including optimal bounds and cross-property bounds, and statistical mechanics. We rigorously prove that anisotropic networks consisting of sets of intersecting parallel channels in the low-density limit, ordered or disordered, possess optimal effective conductivity tensors. For a variety of different disordered networks, we show that when short-range and long-range order increases, there is an increase in both the effective conductivity and elastic moduli of the network. Moreover, we demonstrate that the effective conductivity and elastic moduli of various disordered networks derived from disordered ‘stealthy’ hyperuniform point patterns possess virtually optimal values. We note that the optimal networks for conductivity are also optimal for the fluid permeability associated with slow viscous flow through the channels as well as the mean survival time associated with diffusion-controlled reactions in the channels. In summary, we have identified ordered and disordered hyperuniform low-weight cellular networks that are multifunctional with respect to transport (e.g., heat dissipation and fluid transport), mechanical and electromagnetic properties, which can be readily fabricated using 3D printing and lithographic technologies.