An elastic composite that contains an elastic film laminated to one or more nonwoven materials is provided. By selectively controlling the constituents of the composite and the method by which it is formed, the present inventors have discovered that the composite may be resistant to oils and other skin care compositions. More specifically, the elastic film is formed from one or more low density semi-crystalline propylene-based polymers that possess a unique combination of mechanical, elastic, and oil-resistant properties. By using such elastomeric polymers, the resulting elastic film may also be thermally bonded to the nonwoven web material(s) without the need for oil-resistant adhesives. Further, the bonding conditions may be selected so that apertures do not form in the film. In this manner, the resulting composite is impervious to oils or other liquids (e.g., solvents, water, solutions, etc.) and thus acts as a barrier during use.