  The density-matrix renormalization-group method is used to study the
electronic states of a two-chain Hubbard model for CuO double chains of
PrBa$_2$Cu$_4$O$_8$. We show that the model at quarter filling has the charge
ordered phases with stripe-type and in-line--type patterns in the parameter
space, and in-between, there appears a wide region of vanishing charge gap; the
latter phase is characteristic of either Tomonaga-Luttinger liquid or a
metallic state with a spin gap. We argue that the low-energy electronic state
of the CuO double chains of PrBa$_2$Cu$_4$O$_8$ should be in the metallic state
with a possibly small spin gap.
