As one of the typical organic cathode materials for sodium ion batteries (SIBs), 3,4,9,10-perylenetetracarboxylic dianhydride (PTCDA) is formidably challenged in practical applications because of its low electrical conductivity, sluggish reaction kinetics, inferior rate capability and cycle life. Inspired by the ultra-strong capillarity of wood's aligned hierarchical microchannels toward ions and water during metabolism, we have designed a novel composite nanofibrous organic cathode consisting of PTCDA/nitrogen-doped carbon/carbon nanotubes (PTCDA/NC/CNT) for SIBs. The PTCDA/NC/CNT cathode exhibits rapid ionic/electronic transport properties and ultrafast reaction kinetics owing to the synergistic effects of the interconnected conductive frameworks and ultra-strong capillarity deriving from the hierarchical micro/nanotunnels. As a result, a highly reversible capacity of 135.6 mA h g^-1 at 50 mA g^-1, excellent rate performance and ultra-long cyclic stability with over 95% capacity retention after 500 cycles at 1000 mA g^-1 are achieved for the PTCDA/NC/CNT cathode in SIBs. Remarkably, an all-organic battery using the PTCDA/NC/CNT cathode and conjugated sodium carboxylate/CNT anode also delivers a high energy density of 85 W h kg^-1 at a power density of 665 W kg^-1. This bio-inspired design provides a promising strategy for the development of next-generation all-organic sodium-ion full batteries.