You won't get persistence when you restart the process, but you'll still be able to run code that needs to read and write files.
Try it for yourself: visit https://ktock.github.io/container2wasm-demo/amd64-debian-was... to start a shell against a Debian container, then use "echo 'hi' > hello.txt" to write a file and "cat hello.txt" to read it.
Since it's Linux and you can give it "network access" (using workarounds such as redirecting via a server), you can just use one of the existing ways to mount a remote filesystem that could be stored on the server (such as NFS, sshfs, ...).
Or bridge/mount to the local browser and store it in indexeddb or whatever.
It already has directory mapping when running outside a browser - maybe there's even an easy way to do this more directly than funneling everything through the virtual network.