Using the results of a high-resolution, cosmological hydrodynamical re-simulation of a supercluster-like region we investigate the physical properties of the gas located along the filaments and bridges which constitute the so-called cosmic web. First we analyze the main characteristics of the density, temperature and velocity fields, which have quite different distributions, reflecting the complex dynamics of the structure formation process. Then we quantify the signals which originate from the matter in the filaments by considering different observables. Inside the cosmic web, we find that the halo density is about 4-6 times larger than in the neighbouring region; the bremsstrahlung X-ray surface brightness reaches at most 10e-16 erg/s/cm^2/armin^2; the Compton-y parameter due to the thermal Sunyaev-Zel'dovich effect is about 10e-6; the reduced shear produced by the weak lensing effect is ~ 0.01-0.02. These results confirm the difficulty of an observational detection of the cosmic web. Finally we find that projection effects of the filamentary network can affect the estimates of the properties of single clusters, increasing their X-ray luminosity by less than 10% and their central Compton-y parameter by up to 30% cent.