We use HI absorption measurements to constrain the amount of cool ($\approx 10^4$ K), photoionized gas in the circumgalactic medium (CGM) of dwarf galaxies with $M_* = 10^{6.5-9.5}~M_\odot$ in the nearby Universe ($z<0.3$). We show analytically that volume-filling gas gives an upper limit on the gas mass needed to reproduce a given HI column density profile. We introduce a power-law density profile for the gas distribution and fit our model to archival HI observations to infer the cool CGM gas mass, $M_{\rm cCGM}$, as a function of halo mass. For volume-filling ($f_V=1$) models, we find $M_{\rm cCGM} = 5 \times 10^8-2 \times 10^9~M_{\odot}$, constituting $\lesssim 10\%$ of the halo baryon budget. For clumpy gas, with $f_V=0.01$, the masses are a factor of $\approx 11$ lower, in agreement with our analytic approximation. Our assumption that the measured HI forms entirely in the cool CGM provides a conservative upper limit on $M_{\rm cCGM}$, and possible contributions from the intergalactic medium or warm/hot CGM will further strengthen our result. We estimate the mass uncertainties due to the range of redshifts in our sample and the unknown gas metallicity to be $\approx 15\%$ and $\approx 10\%$, respectively. Our results show that dwarf galaxies have only $\lesssim 15\%$ of their baryon budget in stars and the cool CGM, with the rest residing in the warm/hot CGM or ejected from the dark matter halos.
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