It’s certainly possible to store continuations. You just store the program counter and a copy of the stack at the point where the continuation was captured. (There may be more efficient representations - I’m just talking conceptually here.)

If you google ‘serializable continuations’ you’ll find a reasonable amount of prior art (e.g. http://wiki.call-cc.org/eggref/5/s11n). I think there has never been a good solution to versioning then against changes to the code.