This work concerns the interaction of light membrane structures enclosing incompressible fluids. Large displacements and collapsed boundaries (initially slender subdomains) are characteristic of this class of problems. A finite-element/boundary-element (FE/BE) coupled discretization is discussed in this work which addresses these challenges. Most common linear fluid models have a boundary-integral representation, restricting the problem to the boundary and making them amenable to a boundary element method (BEM) discretization. A marked advantage of this representation with respect to the conventional partial-differential-equation-based view of finite-element methods for fluids is the bypassing of volumetric meshing. The FE/BE approach is therefore an enabling method for fluid-structure interaction (FSI) problems involving membranes, and the present work serves as a proof of concept for this approach. Numerical experiments show the capabilities of the proposed scheme.