[en] We present new upper limits on the neutral hydrogen (H I) 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 survey and Galactic All-Sky Survey data. All of the limits M_{H I}^lim are more stringent than previously reported values, and those from the GBT improve upon constraints in the literature by a median factor of 23. Normalizing by V-band luminosity L_V and dynamical mass M _dyn, we find M_{H I}^lim/L_V∼10⁻³ M_⊙/L_⊙ and M_{H I}^lim/M_dyn∼5×10⁻⁵, irrespective of location in the Galactic halo. Comparing these relative H I 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 H I upper limits therefore provide the clearest picture yet of the environmental dependence of the H I content in Local Volume dwarfs. If ram pressure stripping explains the dearth of H I 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 M_{H I}^lim 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