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This article considers a problem of evaluating barrier option prices when the underlying dynamics are driven by stochastic elasticity of variance (SEV). We employ asymptotic expansions and Mellin transform to evaluate the option prices. The approach is able to efficiently handle barrier options in a SEV framework and produce explicitly a semi-closed form formula for the approximate barrier option prices. The formula is an expansion of the option price in powers of the characteristic amplitude scale and variation time of the elasticity and it can be calculated easily by taking the derivatives of the Black-Scholes price for a barrier option with respect to the underlying price and computing the one-dimensional integrals of some linear combinations of the Greeks with respect to time. We confirm the accuracy of our formula via Monte-Carlo simulation and find the SEV effect on the Black-Scholes barrier option prices.