Metal-organic frameworks (MOF), potentially porous coordination structures, are envisioned for adsorption-based natural gas (ANG) storage, including mobile applications. The factors affecting the performance of the ANG system with a zirconium-based MOF with benzene dicarboxylic acid as a linker (ZrBDC) as an adsorbent were considered: textural properties of the adsorbent and thermal effect arising upon adsorption. The high-density ZrBDC-based pellets were prepared by mechanical compaction of the as-synthesized MOF powder at different pressures from 30 to 240 MPa at 298 K without a binder and mixed with polymer binders: polyvinyl alcohol (PVA) and carboxyl methylcellulose (CMC). The structural investigations revealed that the compaction of ZrBDC with PVA under 30 MPa was optimal to produce the ZrBDC-PVA adsorbent with more than a twofold increase in the packing density and the lowest degradation of the porous structure. The specific total and deliverable volumetric methane storage capacities of the ZrBDC-based adsorbents were evaluated from the experimental data on methane adsorption measured up to 10 MPa and within a temperature range from 253 to 333 K. It was measured experimentally that at 253 K, an 100 mL adsorption tank loaded with the ZrBDC-PVA pellets exhibited the deliverable methane storage capacity of 172 m3(NTP)/m3 when the pressure dropped from 10 to 0.1 MPa. The methane adsorption data for the ZrBDC powder and ZrBDC-PVA pellets were used to calculate the important thermodynamic characteristic of the ZrBDC/CH4 adsorption system—the differential molar isosteric heat of adsorption, which was used to evaluate the state thermodynamic functions: entropy, enthalpy, and heat capacity. The initial heats of methane adsorption in powdered ZrBDC evaluated from the experimental adsorption isosteres were found to be ~19.3 kJ/mol, and then these values in the ZrBDC/CH4 system decreased at different rates during adsorption. In contrast, the heat of methane adsorption onto the ZrBDC-PVA pellets increased from 19.4 kJ/mol to a maximum with a magnitude, width, and position depended on temperature, and then it fell. The behaviors of the thermodynamic state functions of the ZrBDC/CH4 adsorption system were interpreted as a variation in the state of adsorbed molecules determined by a ratio of CH4-CH4 and CH4-ZrBDC interactions. The heat of adsorption was used to calculate the temperature changes of the ANG systems loaded with ZrBDC powder and ZrBDC pellets during methane adsorption under adiabatic conditions; the maximum integrated heat of adsorption was found at 273 K. The maximum temperature changes of the ANG system with the ZrBDC materials during the adsorption (charging) process did not exceed 14 K that are much lower than those reported for the systems loaded with activated carbons. The results obtained are of direct relevance for designing the adsorption-based methane storage systems for the automotive industry, developing new gas-power robotics systems and uncrewed aerial vehicles.