Eleven fluorite oxides with five principal cations (in addition to a four-principal-cation (Hf0.25Zr0.25Ce0.25Y0.25)O2-δ as a start point and baseline) were fabricated via high-energy ball milling, spark plasma sintering, and annealing in air. Eight of the compositions, namely (Hf0.25Zr0.25Ce0.25Y0.25)O2-δ, (Hf0.25Zr0.25Ce0.25)(Y0.125Yb0.125)O2-δ, (Hf0.2Zr0.2Ce0.2)(Y0.2Yb0.2)O2-δ, (Hf0.25Zr0.25Ce0.25)(Y0.125Ca0.125)O2-δ, (Hf0.25Zr0.25Ce0.25)(Y0.125Gd0.125)O2-δ, (Hf0.2Zr0.2Ce0.2)(Y0.2Gd0.2)O2-δ, (Hf0.25Zr0.25Ce0.25)(Yb0.125Gd0.125)O2-δ, and (Hf0.2Zr0.2Ce0.2)(Yb0.2Gd0.2)O2-δ, possess single-phase solid solutions of the fluorite crystal structure with high configurational entropies (on the cation sublattices), akin to those high-entropy alloys and ceramics reported in prior studies. Most high-entropy fluorite oxides (HEFOs), except for the two containing both Yb and Gd, can be sintered to high relative densities. These single-phase HEFOs exhibit lower electrical conductivities and comparable hardness (even with higher contents of softer components such as Y2O3 and Yb2O3), in comparison with 8 mol. % Y2O3-stabilized ZrO2 (8YSZ). Notably, these single-phase HEFOs possess lower thermal conductivities than that of 8YSZ, presumably due to high phonon scattering by multiple cations and strained lattices.