This is why I decided to find something that creates a file on the user's filesystem which the user can find and edit it themselves, and if any changes are made, sync that back up to the extension (either by periodically checking or by using some listener).
I have looked around but nothing really seems to fit what I'm trying to do. Chrome's fileSystem API only works for chrome apps, not chrome extensions and the HTML5 fileSystem API does not allow for a simple filesystem URL to be requested and opened, instead it obfuscates the stored file and makes it practically impossible to edit that file easily.
Is there any way to simply ask for a location to store a file and then allow the user to edit that file and sync it back up?
No, extensions are sandboxed from the real filesystem.
As you said, it's possible to read extension's own files; however, this is read-only for the extension and modifying those files on a deployed extension will result in Chrome detecting extension "compromise" and immediate disabling.
The only way for a Chrome extension to escape the sandbox is, as wOxxOm suggested, a Native Host module. Note that this cannot be distributed in Chrome Web Store with the extension; it needs a separate installer.
Alternatively, you could use some sort of cloud storage with API to access it; e.g. a user could store something in a Dropbox subfolder, and your extension can authorize access to it via Dropbox API. Unfortunately, there is no "native" solution like
syncFileSystem for Apps.