Elastic sheets offer a path to encapsulating a droplet of one fluid in another that is different from that of traditional molecular or particulate surfactants(1). In wrappings of fluids by sheets of moderate thickness with petals designed to curl into closed shapes(2,3), capillarity balances bending forces. Here, we show that, by using much thinner sheets, the constraints of this balance can be lifted to access a regime of high sheet bendability that brings three major advantages: ultrathin sheets automatically achieve optimally efficient shapes that maximize the enclosed volume of liquid for a fixed area of sheet; interfacial energies and mechanical properties of the sheet are irrelevant within this regime, thus allowing for further functionality; and complete coverage of the fluid can be achieved without special sheet designs. We propose and validate a general geometric model that captures the entire range of this new class of wrapped and partially wrapped shapes.