  Interplay of Fermi surface topology and electron correlation is the
quintessential ingredient underlying spontaneous symmetry breaking in itinerant
electronic systems. In one-dimensional (1D) systems at half-filling, the
inherent Fermi surface nesting makes the translationally invariant metallic
state unstable, which is known as Peierls instability. Extending the scope of
Peierls instability to two (2D) or three dimensions (3D), however, is not
straightforward, since the Fermi surface in higher dimensions is generally not
nested. In this work, we show that a perfectly nested Fermi surface can be
realized in a class of 2D perovskite oxides, giving rise to 2D Peierls
instability. Here the central role is played by the zone boundary Dirac line
node (DLN) protected by two orthogonal glide mirrors induced by the rotation of
oxygen octahedra. Especially, at a critical angle of the octahedron rotation,
the zone-boundary DLN flattens, leading to logarithmically diverging
susceptibility. We propose the 2D Peierls instability driven by dispersionless
DLN as a principle mechanism for spontaneous symmetry breaking in various
layered perovskite oxides including the antiferromagnetism of Sr$_2$IrO$_4$. As
a clear signature of the 2D Peierls instability, we predict that the magnetic
domain wall in Sr$_2$IrO$_4$ hosts localized soliton modes.
