Multilayer film webs having a layer of an oxygen-permeable composition that is a blend of i) an organic acid or organic acid salt, ii) a copolymer of a) ethylene, b) an C.sub.3 to C.sub.8 .alpha.,.beta.-ethylenically unsaturated carboxylic acid, and optionally c) an alkyl acrylate or alkyl methacrylate comonomer, where greater than 70% of the total acid moieties of (i) plus (ii) are neutralized to the corresponding metal salts, and iii) at least one additional ethylene-containing polymer selected from the group consisting of ethylene homopolymers and ethylene copolymers other than ethylene acid copolymers and ionomers are used to form packaging materials in a process in which a packaging structure is provided that includes the multilayer film web, a vacuum is applied to the packaging structure, and the perimeters of the film web layers are heat sealed.