Two-dimensional layered materials, particularly ternary chalcogenides, are promising for technological applications, as they exhibit tunable electronic, optical, and magnetic properties. Here we present detailed first-principles calculations of electronic, optical, and thermal transport properties of ${\text{Fe}X}_{2}{Y}_{4}$ $(X=\mathrm{Ga}, \mathrm{In}; Y=\mathrm{S}, \mathrm{Se}, \mathrm{Te})$ ternary layered chalcogenides, relevant for sustainable energy applications. We show that the monolayers, similar to bulk, are dynamically and thermally stable, as demonstrated by phonon dispersion and molecular dynamics simulations. The exfoliation energy of monolayers is comparable to graphene and transition-metal dichalcogenides, suggesting possible synthesis of monolayers by mechanical exfoliation. In the hexagonal crystal structure, these materials are nonmagnetic semiconductors with varied band-gap values, that are interesting for photocatalysis, photovoltaics, and thermoelectric applications. ${\mathrm{FeGa}}_{2}{\mathrm{S}}_{4}$ and ${\mathrm{FeIn}}_{2}{\mathrm{S}}_{4}$ monolayers exhibit suitable band gaps and band-edge positions for photocatalytic water splitting and ${\mathrm{CO}}_{2}$ reduction. ${\mathrm{FeGa}}_{2}{\mathrm{Se}}_{4}$ and ${\mathrm{FeIn}}_{2}{\mathrm{Se}}_{4}$ monolayers are promising photocatalysts for hydrogen evolution reaction. The calculated optical absorption spectra indicate that ${\mathrm{FeIn}}_{2}{\mathrm{Te}}_{4}$, ${\mathrm{FeIn}}_{2}{\mathrm{Se}}_{4}$, and ${\mathrm{FeGa}}_{2}{\mathrm{Se}}_{4}$ are promising photovoltaic absorbers with high spectroscopic limited maximum efficiencies of $\ensuremath{\sim}$ 30%, 27%, and 24.5%, respectively, comparable to existing high-performance thin-film absorber materials, such as CdTe $(\ensuremath{\sim}31.5%)$ and ${\mathrm{CuInSe}}_{2}$ $(\ensuremath{\sim}28%)$. The narrow band gaps and relatively low room-temperature lattice thermal conductivity of ${\mathrm{FeGa}}_{2}{\mathrm{Te}}_{4}$ (11.00 W/mK) and ${\mathrm{FeIn}}_{2}{\mathrm{Te}}_{4}$ (8.84 W/mK) monolayers suggest potential applications in thermoelectrics. We thereby propose ${\text{Fe}X}_{2}{Y}_{4}$ ternary layered chalcogenides as promising materials for sustainable energy applications due to their suitable electronic, optical, and thermal properties.