  First principles investigations of the high temperature superconducting
system Ba$_2$CuO$_{3+\delta}$, recently discovered at $\delta\approx0.2$ at
$T_c=70$ K, are applied to demonstrate the effects of oxygen ordering on the
electronic and magnetic properties. The observed `highly over-doped'
superconducting phase displays stretched Cu-planar oxygen O$_{\rm P}$ distances
and anomalously shortened Cu-apical O$_{\rm A}$ separations compared with other
cuprates. The stoichiometric system $\delta=0$, with its strongly
one-dimensional (1D) Cu-O$_{\rm P}$ chain structure, when nonmagnetic shows 1D
Fermi surfaces that lead, within density functional theory, to
antiferromagnetic Cu-O$_{\rm P}$ chains (a spin-Peierls instability).
Accounting for 1D fluctuations and small interchain coupling according to the
theory of Schulz indicates this system, like Sr$_2$CuO$_3$, is near the 1D
Luttinger-liquid quantum critical phase. The unusual Cu-O bond lengths per se
have limited effects on other properties for $\delta$=0. We find that a
`doubled bilayer' structure of alternating Cu-O$_{\rm P}$ chains and wide rung
Cu$_3$O$_4$ ladders is the energetically preferred one of three possibilities
where the additional oxygen ions bridge Cu-O$_{\rm P}$ chains in the
superconducting phase $\delta=1/4$. Nominal formal valences of the three Cu
sites are discussed. The six-fold (octahedral) site is the most highly
oxidized, accepting somewhat more holes in the $d_{z^2}$ orbital than in the
$d_{x^2-y^2}$ orbital. The implication is that two-band physics is involved in
the pairing mechanism and the superconducting carriers. The Fermi surfaces of
this metallic bilayer structure show both 1D and 2D strong (incipient) nesting
instabilities, possibly accounting for the lack of clean single-phase samples
based on this structure and suggesting importance for the pairing mechanism.
