The first study of the local interstellar medium (LISM) toward Puppis-Vela (1 = 245 degrees to 275 degrees, b = -15 degrees to +5 degrees, d <200 pc) is presented in this paper. A study of the locations, sizes, and physical characteristics of local interstellar gas, i.e.astronephography, is included, and relies upon the improved distance measurements provided by Hipparcos parallax measurements. All spectra of more distant sight lines contain absorption features due to intervening local gas, and more distant structures can only be studied accurately if components due to the LISM have been isolated. Towards this end, high-resolution (R 95,000), high signal-to-noise (S/N similar to 110 to 250) Na I lambda lambda 5889.951, 5895.924 spectra of 11 nearby stars in the direction of Puppis-Vela have been obtained with the Coude Echelle Spectrograph on the 1.4 m Coude Auxiliary Telescope at the European Southern Observatory. Toward Puppis-Vela, absorption due to the Local Interstellar Cloud (LIC) was not observed, but components at three distinct velocities were found, and the extent of the local gas producing the features was estimated. The three components have the following locations and velocities: component A-(l approximate to 276 degrees to 298 degrees, b approximate to -5 degrees to +4 degrees), V-hello = +6 to +9 km s(-1) and d similar to 104 pc; component B-(l approximate to 264 degrees to 276 degrees, b approximate to -7 degrees to + 3 degrees), V-hello = + 12 to +15 km s(-1), and d similar to 115 pc; component C-(l approximate to 252 degrees to 271 degrees, b approximate to -8 degrees to -6 degrees), V-hello = +21 to +23 km s(-1) and d similar to 131 pc. The conclusions regarding the ultraviolet spectrum of gamma (2) Vel (l = 263 degrees, b = -8 degrees, d = 258 +/- 35 pc) presented by Fitzpatrick & Spitzer were reexamined in light of this new LISM data, and the ambiguity in their conclusions about several absorption components is resolved. The stars in Puppis-Vela flank the region of the apparent extension of the Local Bubble (or Cavity) known as the beta CMa tunnel, and measurements of the Na I column density towards the sample stars have been used to modify existing estimates of the extent of the tunnel. A compilation of all existing Na I observations of less than 200 pc sight lines around the tunnel reveal that low column densities have been exclusively detected within l approximate to 210 degrees to 250 degrees, and b approximate to -21 degrees to -9 degrees. Near the Galactic plane, at latitudes - 10 degrees <b <0 degrees and d less than or similar to 150 pc, the tunnel is cofined to l <270, a lower longitude than was previously reported.