We report a detailed spectral analysis of the population of low-mass X-ray binaries (LMXBs) detected in the elliptical galaxy NGC~4278 with Chandra. Seven luminous sources were studied individually, four in globular clusters (GCs), and three in the stellar field. The range of (0.3-8 keV) $L_X$ for these sources suggests that they may be black hole binaries (BHBs). Comparison of our results with simulations allows us to discriminate between disk and power-law dominated emission, pointing to spectral/luminosity variability, reminiscent of Galactic BHBs. The BH masses derived from a comparison of our spectral results with the $L_X sim T^4_{in}$ relation of Galactic BHBs are in the 5-15 $M_{odot}$ range, as observed in the Milky Way. The analysis of joint spectra of sources selected in three luminosity ranges suggests that while the high luminosity sources have prominent thermal disk emission components, power-law components are likely to be important in the mid and low-luminosity spectra. Comparing low-luminosity average spectra, we find a relatively larger $N_H$ in the GC spectrum; we speculate that this may point to either a metallicity effect, or to intrinsic physical differences between field and GC accreting binaries. Analysis of average sample properties uncover a previously unreported $L_X - R_G$ correlation (where $R_G$ is the galactocentric radius) in the GC-LMXB sample, implying richer LMXB populations in more central GCs. No such trend is seen in the field LMXB sample. We can exclude that the GC $L_X - R_G$ correlation is the by-product of a luminosity effect, and suggest that it may be related to the presence of more compact GCs at smaller galactocentric radii, fostering more efficient binary formation.
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