The developer console can be considered part of the application in this regard. Anything you could do in code can be done on the console as well. If your app can access something, so can the user.
And it's even worse than that. Even if your user doesn't care about client-side stores, because it's his data anyway, if your app is vulnerable to XSS, a malicious user exploiting it can also access any data in client-side stores.
In addition to that, if you store something on the client, it may get written to disk (cookies, websql, and so on). So if the attacker has access to the client PC even outside the context of the application, he will also be able to access such data.
So in short, simply don't persist sensitive data on the client.