Does Azure Functions offer a local (state) storage to eliminate a need in invoking additional services such as storage, docDB, etc?
As per David Ebbo's response, Azure Functions are built on the main Azure serverless platform which is App Service.
This link will explain more but I've copied some key pieces in case the link rots.
https://github.com/projectkudu/kudu/wiki/Understanding-the-Azure-App-Service-file-system
Persisted files are those of your deployed program and they are rooted in %HOME% or /home on Linux/Containers. These files are shared between scaled-out instances. You get 1GB.
Temp files can be written to %APPDATA% or %TMP%. These are not shared and they're volatile within restarts of your web app. You get 500MB.
Kudu is the name of the web portal tooling to let you get at the Web App environment but apparently you won't see temp files you've written unless you turn off SCM separation, see link.