The high-redshift radio galaxy MRC 1138−262 ('Spiderweb Galaxy'; z = 2.16) is one of the most massive systems in the early Universe and surrounded by a dense 'web' of proto-cluster galaxies. Using the Australia Telescope Compact Array, we detected CO(1-0) emission from cold molecular gas - the raw ingredient for star formation - across the Spiderweb Galaxy. We infer a molecular gas mass of M H2 = 6 × 1010 M (for M H2/L′CO = 0.8). While the bulk of the molecular gas coincides with the central radio galaxy, there are indications that a substantial fraction of this gas is associated with satellite galaxies or spread across the intergalactic medium on scales of tens of kpc. In addition, we tentatively detect CO(1-0) in the star-forming proto-cluster galaxy HAE 229, 250 kpc to the West. Our observations are consistent with the fact that the Spiderweb Galaxy is building up its stellar mass through a massive burst of widespread star formation. At maximum star formation efficiency, the molecular gas will be able to sustain the current star formation rate (SFR 1400 M yr−1, as traced by Seymour et al.) for about 40 Myr. This is similar to the estimated typical lifetime of a major starburst event in infrared luminous merger systems.