We have utilized molecular-beam-epitaxy techniques in an electron-cyclotron-resonance oxygen plasma to synthesize thin films of NiO, Fe3O4 (magnetite), and modulated structures of these compounds with repeat structures of (17 angstrom)/(17 angstrom), (34 angstrom)/(34 angstrom), and (68 angstrom)/(68 angstrom). The current-voltage (I-V) relation has been measured for current parallel and perpendicular to the planes, for temperature from 5 to 300 K, and with electric fields E up to 10(5) V/cm. All samples show strong nonlinear effects in the I-V behavior with I/V proportional to exp(cE1/2). For NiO and Fe3O4, c is approximately proportional to 1/T below 100 K, as expected for Poole-Frenkel conduction and, on the basis of this interpretation, yields dielectric constants of 4.1 and 10.1, respectively. A nonmonotonic temperature dependence of c occurs for the modulated structures with a minimum in the field dependence occurring at 40 K. The low-field conductivities found for pure NiO and Fe3O4 films are similar to those reported for the bulk at the higher temperatures, with Fe3O4 films also showing the Verwey transition at 120 K. In the (34 angstrom)/(34 angstrom) modulated structure, the electrical conductivity and its temperature dependence are consistent with pure Fe3O4 in the low-temperature phase, but the Verwey transition upon warming to the high-conductivity phase does not occur at any T < 300 K, suggesting that low-dimensionality effects may alter the transition. The (34 angstrom)/(34 angstrom) modulated structure also shows a low-E-field conductivity anisotropy of 10(6) or greater, which is comparable to the largest known values for any material. We give arguments why modulated structures may modify the electrical-transport properties of material containing Bohr-like impurity states and show that Fe3O4/NiO may be such a case. We conclude that a modulated and E-field-variable conductivity of many orders of magnitude can be achieved in the modulated structures on a length scale of tens of angstroms, which suggests the possibility of voltage-controlled superlattice phenomena.