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We report results of investigation of the current-carrying capacity of nanowires made from the quasi-1D van der Waals metal tantalum triselenide capped with quasi-2D boron nitride. The chemical vapor transport method followed by chemical and mechanical exfoliation were used to fabricate mm-long TaSe3 wires with lateral dimensions in the 20 to 70 nm range. Electrical measurements establish that TaSe3/h-BN nanowire heterostructures have a breakdown current density exceeding 10 MA/cm2 - an order-of-magnitude higher than that in copper. Some devices exhibited an intriguing step-like breakdown, which can be explained by the atomic thread bundle structure of the nanowires. The quasi-1D single crystal nature of TaSe3 results in low surface roughness and the absence of grain boundaries; these features potentially can enable the downscaling of these wires to lateral dimensions in the few-nm range. These results suggest that quasi-1D van der Waals metals have potential for applications in the ultimately downscaled local interconnects., Comment: 22 pages, 6 figures