A container assembly has a soft-sided insulated wall structure. It has one or more internal chambers in which to place objects to be kept cool or warm. A first closure member governs access to the container assembly. The wall structure includes one or more accommodations in which to place a warming or cooling element, such as an ice pack. The accommodations in the outside walls are externally accessible without opening the main chamber, and have their own closure, and may be insulated. The inside is a thin web through which to transfer heat. An internal partition divides the container assembly into two chambers. The partition may have a thermal storage element. The container may be a collapsible foldable bag; a tote bag; back pack or a non-collapsible lunch box. The thermal storage elements may have a flat disk shape. The disks slide sideways into place in the respective accommodations.