We report polymer composite films containing fillers comprised of quasi-one-dimensional (1D) van der Waals materials, specifically transition metal trichalcogenides containing 1D structural motifs that enable their exfoliation into bundles of atomic threads. These nanostructures are characterized by extremely large aspect ratios of up to 10^6. The polymer composites with low loadings of quasi-1D TaSe3 fillers (below 3 vol. %) revealed excellent electromagnetic interference shielding in the X-band GHz and EHF sub-THz frequency ranges, while remaining DC electrically insulating. The unique electromagnetic shielding characteristics of these films are attributed to effective coupling of the electromagnetic waves to the high-aspect-ratio electrically-conductive TaSe3 atomic-thread bundles even when the filler concentration is below the electrical percolation threshold. These novel films are promising for high-frequency communication technologies, which require electromagnetic shielding films that are flexible, lightweight, corrosion resistant, electrically insulating and inexpensive.