We demonstrate the formation of large sheets of layered organic-inorganic perovskite (OIPC) crystals, as thin as a single unit cell, prepared by mechanical exfoliation. The resulting two-dimensional OIPC nanosheets of 2.4 nm thickness are direct semiconductors with an optical band gap of 2.4 eV. They exhibit unusually strong light-matter interaction with an optical absorption as high as 25% at the main excitonic resonance, as well as bright photoluminescence. We extract an exciton binding energy of 490 meV from measurement of the series of excited exciton states. The properties of the excitons are shown to be strongly influenced by the changes in the dielectric surroundings. The environmental sensitivity of these ultrathin OIPC sheets is further reflected in the strong suppression of a thermally driven phase transition present in the bulk crystals.