> How can you "Make illegal states unrepresentable" with mutable state

By either making the data const, or encapsulating mutable state with private fields and public methods

> How can you do "Errors as values"

Go, Rust, Odin, Zig, and many more are imperative languages that do exactly this

> How can you do "Functional core, imperative shell"

Write a function that takes the old state and returns the new state

> By either marking your data as const, or encapsulating mutations with private fields and public methods

That would seem to be making illegal states unreachable rather than unrepresentable, closer in spirit to "parse, don't verify".