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We examine the gas content of field dwarf galaxies in a high-resolution cosmological simulation. In agreement with previous work, we find that galaxies inhabiting dark matter haloes with mass below a critical value, $M_{200} \lesssim M_{\rm crit} \approx 5\times 10^{9} \ M_{\odot}$, are quiescent at the present day. The gas content of these galaxies is thus insensitive to feedback from evolving stars. Almost half of these quiescent systems today have gas masses much smaller than that expected for their mass. We find that gas-deficient galaxies originate from 1) past interactions with massive hosts, in which a dwarf loses gas and dark matter via tidal and ram-pressure forces; and 2) from hydrodynamic interactions with the gaseous filaments and sheets of the cosmic web, in which a dwarf loses gas via ram-pressure. We refer to these systems as ``flybys'' and ``COSWEBs''. Flybys locate in high-density regions, tracing the location of the most massive galaxies in the simulation. In contrast, COSWEBs are dispersed throughout the volume and trace the cosmic web. For sub-critical systems, $M_{200} < M_{\rm crit}$, the fraction of COSWEB galaxies can be as high as $35 \%$, and much higher for flybys, which make up 100 per cent of the galaxies with $M_{200}<3\times 10^8 \ \rm M_{\odot}$. The deficit of gas caused by these mechanisms may preclude the detection of a large fraction of field dwarfs in future HI surveys. For galaxies inhabiting halos with mass $M_{200} > M_{\rm crit}$, we find that cosmic web stripping, on average, shuts down star formation in more than $70\%$ of the affected systems.