Claude Code for the web is kind of a persistent virtual dev environment already.

You can start a session there and chat with it to get a bunch of work done, then come back to that session a day later and the virtual filesystem is in the same state as when you left it.

I haven't figured out if this has a time limit on it - it's possible they're doing something clever with object storage such that the cost of persisting those environments is really low, see also Fly's Sprites.dev: https://fly.io/blog/design-and-implementation/

It's so incredibly buggy though. I end up with hung sessions "starting claude code" every second or third time. After a few times of losing work I'm done with it. I'll check back in a few months and see if it's in better shape.