This article designs a new fifth-order finite volume mapped unequal-sized weighted essentially non-oscillatory scheme (MUS-WENO) for solving hyperbolic conservation laws on structured meshes. One advantage is that the new mapped WENO-type spatial reconstruction is a convex combination of a quartic polynomial with two linear polynomials defined on unequal-sized central or biased spatial stencils. Then we propose the new mapped nonlinear weights and new mapping function to decrease the difference between the linear weights and nonlinear weights. This method has the characteristics of small truncation errors and high-order accuracy. And it could give optimal fifth-order convergence with a very tiny $\varepsilon$ even near critical points in smooth regions while suppressing spurious oscillations near strong discontinuities. Compared with the classical finite volume WENO schemes and mapped WENO (MWENO) schemes, the linear weights can be any positive numbers on the condition that their summation is one, which greatly reduces the calculation cost. Finally, we propose a new modified positivity-preserving method for solving some low density, low pressure, or low energy problems. Extensive numerical examples including some unsteady-state problems, steady-state problems, and extreme problems are used to testify to the efficiency of this new finite volume MUS-WENO scheme.