(Abridged) Using the Arecibo Observatory we have obtained neutral hydrogen (HI) absorption and emission spectral pairs in the direction of 26 background radio continuum sources in the vicinity of the Perseus molecular cloud. Strong absorption lines were detected in all cases allowing us to estimate spin temperature (T_s) and optical depth for 107 individual Gaussian components along these lines of sight. Basic properties of individual HI clouds (spin temperature, optical depth, and the column density of the cold and warm neutral medium, CNM and WNM) in and around Perseus are very similar to those found for random interstellar lines of sight sampled by the Millennium HI survey. This suggests that the neutral gas found in and around molecular clouds is not atypical. However, lines of sight in the vicinity of Perseus have on average a higher total HI column density and the CNM fraction, suggesting an enhanced amount of cold HI relative to an average interstellar field. Our estimated optical depth and spin temperature are in stark contrast with the recent attempt at using Planck data to estimate properties of the optically thick HI. Only ~15% of lines of sight in our study have a column density weighted average spin temperature lower than 50 K, in comparison with >85% of Plancks sky coverage. The observed CNM fraction is inversely proportional to the optical-depth weighted average spin temperature, in excellent agreement with the recent numerical simulations by Kim et al. While the CNM fraction is on average higher around Perseus relative to a random interstellar field, it is generally low, 10-50%. This suggests that extended WNM envelopes around molecular clouds, and/or significant mixing of CNM and WNM throughout molecular clouds, are present and should be considered in the models of molecule and star formation.
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