Technically possible. There are a few Scheme implementations that have persistent continuation (all I know is JVM based ones). Also this page contains a link to the slides that talks about persistent delimited continuation in OCaml: http://logic.cs.tsukuba.ac.jp/Continuation/program.html
Making continuations serializable may cost its implementation, so whether it scales well for yc.news is anothoer question. I don't know the answer for it.
In lua at least, continuations have a 1k stack overhead in memory per continuation. I believe they are serializable as well... I'll have to double-check that. At 1k+ a pop they're not cheap, but are manageable.
Having my tabs open last for a couple of days instead of 30 minutes would make the site more pleasant for me.
Making continuations serializable may cost its implementation, so whether it scales well for yc.news is anothoer question. I don't know the answer for it.