Topological flat bands (FBs) offer an ideal platform for realizing exotic topological phases and exploring quantum geometric effects, yet their realization with both exact flatness and stable topology in local lattice models has been hindered by fundamental no-go theorems. The obstruction is also manifested as the absence of exact Gaussian TNS representations for topological insulators and superconductors. Here, we overcome this barrier by demonstrating the existence of critical topological FBs (CTFBs) in finite-range hopping models. They saturate the no-go theorems via a unique structure of Bloch wavefunctions: While continuous over the whole Brillouin zone, the projector P(k) onto FBs is non-analytic at isolated band touching points. Filling such CTFBs yields short-range entangled topological states with power-law correlations due to the non-analyticity. We then establish a general symmetry-based principle to systematically construct CTFBs that carry desired topological invariants in given space groups. It utilizes the bipartite structure and requires no further fine-tuning, and the topology is robust against arbitrary symmetry-preserving gap-opening perturbations. Remarkably, independent tuning of the quantum geometry is allowed. Examples exhibiting Chern numbers 1, 2, 3, 6, in 2D, and strong Z₂ indices in 2D and 3D are demonstrated for concreteness. An automated algorithm further identifies more than 50,000 symmetry-indicated CTFBs, including crystalline and higher-order topological ones. In the end, we show that the bipartite structure naturally endows the filled CTFB states with exact TNS representations with finite bond dimensions. By bridging the topological band theory to TNS methods, CTFB provides a novel tractable starting point for exploring strongly correlated topological matter, with potential relevance to realistic material realizations.