We have used the Arecibo Telescope and the Green Bank Telescope to carry out a deep search for H{sc i}~21,cm emission from a large sample of Green Pea galaxies, yielding 19 detections, and 21 upper limits on the H{sc i} mass. We obtain H{sc i} masses of $rm M_{HI} approx (4-300) times 10^8 , rm M_odot$ for the detections, with a median H{sc i} mass of $approx 2.6 times 10^9 , rm M_odot$; for the non-detections, the median $3sigma$ upper limit on the H{sc i} mass is $approx 5.5 times 10^8 , rm M_odot$. These are the first estimates of the atomic gas content of Green Pea galaxies. We find that the H{sc i}-to-stellar mass ratio in Green Peas is consistent with trends identified in star-forming galaxies in the local Universe. However, the median H{sc i} depletion timescale in Green Peas is $approx 0.6$~Gyr, an order of magnitude lower than that obtained in local star-forming galaxies. This implies that Green Peas consume their atomic gas on very short timescales. A significant fraction of the Green Peas of our sample lie $gtrsim 0.6$~dex ($2sigma$) above the local $rm M_{HI} - M_B$ relation, suggesting recent gas accretion. Further, $approx 30$% of the Green Peas are more than $pm 2sigma$ deviant from this relation, suggesting possible bimodality in the Green Pea population. We obtain a low H{sc i}~21,cm detection rate in the Green Peas with the highest O32~$equiv$~[O{sc iii}]$lambda$5007/[O{sc ii}]$lambda$3727 luminosity ratios, O32~$> 10$, consistent with the high expected Lyman-continuum leakage from these galaxies.
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