We compute the electronic structure, spin and charge state of Fe ions, and structural phase stability of paramagnetic CaFeO$_3$ under pressure using a fully self-consistent in charge density DFT+dynamical mean-field theory method. We show that at ambient pressure CaFeO$_3$ is a negative charge-transfer insulator characterized by strong localization of the Fe $3d$ electrons. It crystallizes in the monoclinic $P2_1/n$ crystal structure with a cooperative breathing mode distortion of the lattice. While the Fe $3d$ Wannier occupations and local moments are consistent with robust charge disproportionation of Fe ions in the insulating $P2_1/n$ phase, the physical charge density difference around the structurally distinct Fe A and Fe B ions with the ``contracted and ``expanded oxygen octahedra, respectively, is rather weak, $sim$0.04. This implies the importance of the Fe $3d$ and O $2p$ negative charge transfer and supports the formation of a bond-disproportionated state characterized by the Fe A $3d^{5-delta}underline{L}^{2-delta}$ and Fe B $3d^5$ valence configurations with $delta ll 1$, in agreement with strong hybridization between the Fe $3d$ and O $2p$ states. Upon compression above $sim$41 GPa CaFeO$_3$ undergoes the insulator-to-metal phase transition (IMT) which is accompanied by a structural transformation into the orthorhombic $Pbnm$ phase. The phase transition is accompanied by suppression of the cooperative breathing mode distortion of the lattice and, hence, results in the melting of bond disproportionation of the Fe ions. Our analysis suggests that the IMT transition is associated with orbital-dependent delocalization of the Fe $3d$ electrons and leads to a remarkable collapse of the local magnetic moments. Our results imply the crucial importance of the interplay of electronic correlations and structural effects to explain the properties of CaFeO$_3$.
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