We introduce aqueous ionic liquid (IL) mixtures, specifically mixtures of 1-butyl-3-imidazoliumtetrafluoroborate (BMImBF4), with water as a minimal model of lipid bilayer membranes. Imidazolium-based ILs are known to form clustered nanoscale structures in which local inhomogeneities, micellar or lamellar structures, are formed to shield hydrophobic parts of the cation from the polar cosolvent (water). To investigate these nanostructures, dynamic light scattering (DLS) on samples with different mixing ratios of water and BMImBF4 was performed. At mixing ratios of 50% and 45% (v/v), small and homogeneous nanostructures can indeed be detected. To test whether, in particular, these stable nanostructures in aqueous mixtures may mimic the effects of phospholipid bilayer membranes, we further investigated their interaction with myelin basic protein (MBP), a peripheral, intrinsically disordered membrane protein of the myelin sheath. Using dynamic light scattering (DLS), continuous wave (CW) and pulse electron paramagnetic resonance (EPR), and small-angle X-ray scattering (SAXS) on recombinantly produced, "healthy" charge variants rmC1WT and double cysteine variant C1S17CH85C, we find that the size and the shape of the determined nanostructures in an optimum mixture offer model membranes in which the protein exhibits native behavior. SAXS measurements illuminate the size and shape of the nanostructures and indicate IL-rich "beads" clipped together by functional MBP, one of the in vivo roles of the protein in the myelin sheath. All the gathered data combined indicate that the 50% and 45% aqueous IL mixtures can be described as offering minimal models of a lipid mono- or bilayer that allow native processing and potential study of at least peripheral membrane proteins like MBP.