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Aims. We present the X-ray imaging and spectral analysis of the diffuse emission around the radio galaxy J1140-2629 (the Spiderweb galaxy) at z = 2.16 and of its nuclear emission, based on a deep (700 ks) Chandra observation. Methods. We obtained a robust characterization of the unresolved nuclear emission, and carefully computed the contamination in the surrounding regions due to the wings of the instrument point spread function. Then, we quantified the extended emission within a radius of 12 arcsec. We used the Jansky Very Large Array radio image to identify the regions overlapping the jets, and performed X-ray spectral analysis separately in the jet regions and in the complementary area. Results. We find that the Spiderweb galaxy hosts a mildly absorbed quasar, showing a modest yet significant spectral and flux variability on a timescale of ∼1 year (observed frame). We find that the emission in the jet regions is well described by a power law with a spectral index of Γ ∼ 2 − 2.5, and it is consistent with inverse-Compton upscattering of the cosmic microwave background photons by the relativistic electrons. We also find a roughly symmetric, diffuse emission within a radius of ∼100 kpc centered on the Spiderweb galaxy. This emission, which is not associated with the jets, is significantly softer and consistent with thermal bremsstrahlung from a hot intracluster medium (ICM) with a temperature of kT = 2.0−0.4+0.7 k T = 2. 0 − 0.4 + 0.7 $ kT = 2.0_{-0.4}^{+0.7} $ keV, and a metallicity of Z <  1.6 Z⊙ at 1σ c.l. The average electron density within 100 kpc is ne = (1.51  ±  0.24 ± 0.14) × 10−2 cm−3, corresponding to an upper limit for the total ICM mass of ≤(1.76  ±  0.30  ±  0.17) × 1012 M⊙ (where error bars are 1σ statistical and systematic, respectively). The rest-frame luminosity L0.5 − 10 keV = (2.0 ± 0.5) × 1044 erg s−1 is about a factor of 2 higher than the extrapolated L − T relation for massive clusters, but still consistent within the scatter. If we apply hydrostatic equilibrium to the ICM, we measure a total gravitational mass M(<100 kpc) = (1.5−0.3+0.5) × 1013 M⊙ M (< 100 kpc) = (1. 5 − 0.3 + 0.5) × 10 13 M ⊙ $ M({ and, extrapolating at larger radii, we estimate a total mass M500 = (3.2−0.6+1.1) × 1013 M⊙ M 500 = (3. 2 − 0.6 + 1.1) × 10 13 M ⊙ $ M_{500} = (3.2^{+1.1}_{-0.6})\times 10^{13}\, M_\odot $ within a radius of r500 = (220 ± 30) kpc. Conclusions. We conclude that the Spiderweb protocluster shows significant diffuse emission within a radius of 12 arcsec, whose major contribution is provided by inverse-Compton scattering associated with the radio jets. Outside the jet regions, we also identified thermal emission within a radius of ∼100 kpc, revealing the presence of hot, diffuse baryons that may represent the embryonic virialized halo of the forming cluster. [ABSTRACT FROM AUTHOR]