This paper reports very detailed low-temperature vacuum tunneling spectroscopy investigations of Bi2Sr2CaCu2O8+δ (BSCCO) single crystals. For clean vacuum junctions formed between a cleaved BSCCO single crystal and the normal-metal tip of a scanning tunneling microscope, we obtain stable c-axis vacuum tunneling conditions that allow very reproducible low-temperature electron tunneling spectroscopy. A vacuum junction is identified by tunneling spectra which neither depend on tip/sample spacing nor change as a function of time and position on the sample in the Meissner state. In contrast to the frequently reported linear or parabolic increase with increasing bias voltage, the background conductance of such spectra is largely constant with a slight decrease up to ±300 meV bias. The normal-state conductance inferred from this background has a local maximum at negative sample bias, indicating a pileup below the Fermi level in the (ab)-plane normal-state density of states of BSCCO. Vacuum junctions at 4.8 K show a well developed superconducting gap with large peaks at the gap edges and a finite density of quasiparticle excitations filling the gap. These characteristics are not consistent with an isotropic BCS-like gap parameter. Outside the superconducting gap, the differential conductance curves are very asymmetric, with a striking dip which appears at negative sample bias only. This dip contributes a substantial amount of states to satisfy the conservation of states we find between normal and superconducting BSCCO.