Abstract: Containers and methods are disclosed whereby portions of an available stand-by supply of flowable material within a container may be dispensed substantially immediately on-demand. Preferably, the containers include an open-ended container shell having a resilient, shape-retaining side wall defining an interior space, a liner positioned within the interior space of the container shell, and a cap closing the open-ended container shell to seal the liner against leakage of flowable material. The liner has a flaccid liner body for containing a flowable material to be dispensed. The cap has a discharge opening to allow an amount of flowable material to be discharged therethrough from the liner body. Importantly, the container includes a valve system which includes one-way discharge and inlet valves.