The question whether two liquids mix or not is mainly to do with the intermolecular forces, not the densities. These forces lead to the bulk property known as chemical potential.
Assuming the experiment is done in a pressurized cabin, then the ambient temperature and pressure may take ordinary values such as STP (room temperature, 1 atmosphere of pressure).
At such ordinary temperature and pressure, what happens with oil and water is that the fluid consists of two parts, one of which is mostly (but not completely) water and the other is mostly (but not completely) oil. The entropy of system-plus-environment is maximised when the Gibbs function is minimised, with the result that the oil is not fully dissolved but rather the liquid consists of these two parts. The two parts will gather at separate locations so as to minimise any boundary areas where there is surface tension, and also in response to any ambient forces such as gravity.
In the absence of gravity, therefore, one expects a blob of one liquid and a blob of the other. There are three types of boundary: water-oil, water-air, oil-air, each with a different surface tension. I think the water-air boundary has the highest surface tension, which suggests the oil will surround the water so as to elliminate this boundary. Therefore if there is enough oil then I expect the equilibrium configuration is a sphere of water surrounded by a spherical shell of oil. At smaller amounts of oil I guess the oil will be spread out over the surface of the water, not quite covering it.
Perhaps the configuration with the least Gibbs potential is some other shape (depending on the proportions of oil and water), but the equilibrium configuration will certainly consist of two parts with different concentrations as I have said, unless the temperature is high. At high temperature the two fluids fully dissolve into one another and then you would have a single continuous substance, not two.