We present new upper limits on the neutral hydrogen (HI) content within the stellar half-light ellipses of 15 Galactic dwarf spheroidal galaxies (dSphs), derived from pointed observations with the Green Bank Telescope (GBT) as well as Arecibo L-band Fast ALFA (ALFALFA) survey and Galactic All-Sky Survey (GASS) data. All of the limits Mlim are more stringent than previously reported values, and those from the GBT improve upon contraints in the literature by a median factor of 23. Normalizing by V-band luminosity Lv and dynamical mass Mdyn, we find Mlim/Lv ~ 10^{-3} Mo/Lo and Mlim/Mdyn ~ 5 x 10^{-5}, irrespective of location in the Galactic halo. Comparing these relative HI contents to those of the Local Group and nearby neighbor dwarfs compiled by McConnachie, we find that the Galactic dSphs are extremely gas-poor. Our HI upper limits therefore provide the clearest picture yet of the environmental dependence of the HI content in Local Volume dwarfs. If ram pressure stripping explains the dearth of HI in these systems, then orbits in a relatively massive Milky Way are favored for the outer halo dSph Leo I, while Leo II and Canes Venatici I have had a pericentric passage in the past. For Draco and Ursa Minor, the interstellar medium mass that should accumulate through stellar mass loss in between pericentric passages exceeds Mlim by a factor of ~30. In Ursa Minor, this implies that either this material is not in the atomic phase, or that another mechanism clears the recycled gas on shorter timescales.
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