Electrostatic dissipative (ESD), anti-static (AS), and electromagnetic interference (EMI) shielding materials are commonly based on commodity fossil-fuel-based plastics. This, in turn, contributes to ever-growing non-biodegradable plastic pollution. Graphene nanoplatelets (GN), multi-walled carbon nanotubes (MWCNT), nanostructured carbon black (NCB), and amorphous carbon black (CB) were utilized as nanofillers to prepare bio-based and biodegradable poly(butylene succinate-co-adipate) (PBSA) nanocomposites. Solvent-cast composites were prepared with 1.1 to 30.0 vol.% nanoparticle loading. The literature mainly focuses on relatively low loadings; therefore, for this research, filler loadings were increased up to 30 vol.% but the maximum loading for NCB and CB loadings only reached 17.4 vol.% due to a lack of dimensional stability at higher loadings. The composites were characterized using tensile testing, volumetric and surface conductivity measurements, thermal conductivity measurements, dielectric spectroscopy in the microwave region, and transmittance in the terahertz range. Tensile tests showed excellent carbon filler compatibility and enhanced tensile strength for loadings up to 5 vol.% (up to 20 vol.% for MWCNT). The highest thermal conductivity values were reached for the MWCNT filler, with the 30.0 vol.% filled composite reaching 0.756 W/mK (262% increase over PBSA). All fillers were able to produce composites that yielded volume conductivities above 10−10 S/m. Composites with MWCNT, GN, and NCB inclusions above the percolation threshold are suitable for EMI applications in the microwave and THz frequency range.