Structural instability and limited operational efficiency of sodium-ion battery (SIB) anodes, compared lithium-ion battery (LIB) anodes, are main hurdles for their large-scale applications that can be tackled using self-tunable electrode materials. Here, we created and utilized ultrathin partially-crystalline carbon nanocups (CNCs) that have the ability to self-tune their structure, contain highly tangled walls and present open framework for enhanced energy densities for SIBs and the outcomes are benchmarked with LIBs. The tangled wall structure offers enhanced spacings for ionic storage and the open framework supports enhanced mass transport, while partial-crystallinity of CNCs provides high conductivity and stabilizes the structure for long cycle life. The CNCs delivered high capacities up to 468 mAh/g for SIBs at 25 mA/g and 953 mAh/g for LIBs at 50 mA/g after 100 cycles, respectively, while bore excellent stability by delivering highly stable performance exceeding 1000 cycles for SIBs at 1.5 A/g and 5000 cycles for LIBs at 7.5 A/g (20 C) with only a loss of 0.013 mAh/g per cycle. Moreover, when tested in lower voltage ranges, the CNCs delivered unprecedented capacity of 368 mAh/g (0.01-2.0 V) for SIBs and 671 mAh/g (0.01-2.0 V) and 449 mAh/g (0.01-1.5 V) for LIBs at 25 and 50 mA/g, respectively, which indicate the usability of the CNCs in full-cells. The post-electrochemical characterizations revealed that the developed CNCs self-modified their interlayer spacings (average spacing increased to 0.44 nm from 0.42 nm) during charge-discharge to create additional voids for increased ionic storage and stable performance, especially for Na+, while maintaining their overall morphology, composition and structure. The structural design presented here will provide new pathways of tailoring carbonaceous nanostructures for future high-performance energy storage devices.