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Striking and regular gross structure has been observed in excitation functions for the ${\mathrm{O}}^{16}$+${\mathrm{O}}^{16}$ elastic scattering interaction in the oxygen-ion laboratory energy range from 20 to 80 MeV. Excitation functions have been measured simultaneously at five c.m. angles from 50\ifmmode^\circ\else\textdegree\fi{} to 90\ifmmode^\circ\else\textdegree\fi{}; systematic angular-distribution measurements have been carried out at narrow energy intervals spanning the gross-structure peak in the 90\ifmmode^\circ\else\textdegree\fi{} excitation function for $19\ensuremath{\le}{E}_{\mathrm{c}.\mathrm{m}.}\ensuremath{\le}22$ MeV and at intervals throughout the remainder of the energy range studied. The gross structure is reproduced surprisingly well by a Woods-Saxon optical model having an unusually shallow real well depth ($V=17$ MeV) and an energy-dependent imaginary well depth ($W=0.4 \mathrm{MeV} +0.1 {E}_{\mathrm{c}.\mathrm{m}.}$). Extensive scans have demonstrated that this is the lowest member of a discrete set of equivalent Woods-Saxon potentials; it is the only one of these, however, which provides a reproduction of the experimental data without requiring a marked and complex energy dependence of the real potential-well depth. The effects of adding a repulsive core to this lowest Woods-Saxon potential were studied in some detail; it has been found that the model predictions are quite sensitive to the core---a somewhat surprising result, perhaps reflecting the apparent long mean free path of the ${\mathrm{O}}^{16}$ ions under the present conditions. An ambiguity in the imaginary well depth has been examined in some detail; the present data do not permit resolution of this ambiguity, but throw into question the physical significance of this long mean free path. The addition of a core does not provide significantly improved fits to the experimental data. This does not preclude the the existence of the core, but suggests further study with different well parameters and shapes. A phase-shift analysis of the 28 angular distributions spanning the gross-structure peak provided no evidence for underlying resonances; no compelling explanation for the intermediate (\ensuremath{\sim}200-keV) width structure superposed on the gross-structure peak is yet available. Finer-grained excitation-function structure is present and is attributed to statistical fluctuations in the usual way. Extensive theoretical studies stimulated by, and concurrent with, these experimental studies have demonstrated that these heavy-ion data may provide unique information on the nuclear-matter problem in finite systems, including an experimental determination of the effective nuclear compressibility.