Graph-theoretic topological frustration is a peculiar situation on a finite piece of the honeycomb lattice that prevents a full pairwise coupling of the lattice sites via nearest neighbor links, even when the total number of sites is an even number and the structure possesses no sublattice imbalance. This type of frustration is inherent for organic molecules that are classified as concealed non-Kekulean hydrocarbons, representing peculiar diradicals. Here we show that this topological frustration persists in two-dimensional systems based on the honeycomb lattice. Such systems exhibit fully flat electronic energy bands located at the Fermi level. Therefore, two-dimensional ultimately flat bands can be systematically and predictably constructed for graphene monolayer nanomeshes. These systems are prone to antiferromagnetic ordering and hybrid spin-wave excitations mixing weak ferromagnetic and strong antiferromagnetic features, which could pave the way towards low-power, compact, and ultrafast organic spintronics with near room-temperature operation.