Compositions and packaging films are provided having a multilayer oxygen barrier comprising an EVOH copolymer layer in contact with at least one polyamide layer. The EVOH copolymer compositions preferably comprise about 44 mol %, or less, of ethylene. Multilayer packaging films can optionally include an exterior layer comprising heat resistant materials such as polyester or polyamide. The multilayer oxygen barriers described herein can provide high shrink films with desirably high oxygen barrier properties. The oxygen barriers can be used in making heat-shrinkable films and food packages having desirable high levels of free shrink.