Yeah I caught myself above, we probably cross-posted ...
Continuations do nothing to solve the problem of storing of state. It simply makes the storing of state implicit through the use of language constructs which invoke state-saving and control-flow mechanisms in a way that let you program in a more natural style.
You could, for instance, use our existing session storage and other centralized state-management procedures as the base for implementing continuation semantics in some sort of Tcl+ extension designed to run in our environment.
So that brings me back to my statement above: are web GUIs, at least for the kind of sites you write today and expect to write in the future, complex enough to worry about?