Transition metal chalcogenides have been regarded as promising storage materials for sodium ions owing to their high theoretical capacity. Herein, copper-based metal–organic frameworks (Cu-BTC) are reported as precursors to fabrica copper chalcogenides-carbon composites, namely Cu$_{1.8}$S@C and Cu$_{2-x}$Se@C. The materials exhibit excellent electrochemical performance with high specific capacities (504 mAh g$^{-1}$ for Cu$_{1.8}$S@C and 317 mAh g$^{-1}$ for Cu$_{2-x}$Se@C at 0.1 A g$^{-1}$) and long-term cycling stability when used as anode materials in cells employing carbon-coated Na$_{3}$V$_{2}$(PO$_{4}$)$_{3}$ (NVP/C) positive electrodes. The Cu$_{2-x}$Se@C||NVP/C cell delivers a specific capacity of 73 mAh g$^{-1}$ at 1.2 A g$^{-1}$ (based on cathode mass) and excellent cycling stability (capacity retention of 85% after 500 cycles at 0.12 A g$^{-1}$) with Coulombic efficiency of ≈99.9%. Moreover, the Cu$_{2-x}$Se@C composite performs well as positive electrode storage material in a sodium-metal cell, offering a high reversible capacity of 216 mAh (per gram of Cu$_{2-x}$Se@C) after 1800 cycles at 2 A g$^{-1}$ and enabling high specific energy and power.