There is a pref to define this:

• offline-apps.quota.warn = 51200

You can open the about:config page via the location/address bar. You can accept the warning and click "I'll be careful" to continue.

• http://kb.mozillazine.org/about:config

I don't understand. Are you saying that sites can store data for offline use without asking if it's less bytes than a certain amount or something? I don't get what that pref means or what values it would take and what they mean.

The offline cache is only used in very rare cases to store data locally. Firefox will store small amounts (less than 50 MB) of data in the offline (application) cache without asking for permission.

• offline-apps.allow_by_default = true
• offline-apps.quota.warn = 51200

You can toggle the offline-apps.allow_by_default pref to false to make Firefox ask.

Thanks I will try that.

So, why isn't the checkbox in options titled "tell me when a website asks to store data for offline use" controlling that pref? I guess it's either a bug that it doesn't control it, or a UI bug that it says it like that.

Yes, a website can store data for offline use without asking, and may do so even if you have checked the option "tell me when a website asks to store data for offline use", because that option doesn't work properly. See bug: https://bugzilla.mozilla.org/show_bug.cgi?id=959985

As far as I can tell, it doesn't mean that the site simply "didn't ask", it's just that the option doesn't work (or perhaps only works in cases where the data is larger than pref offline-apps.quota.warn?)

The work-around as given by cor-el above, is to set offline-apps.allow_by_default to false.

The checkbox corresponds to a different preference:

browser.offline-apps.notify

Edit: I missed your last reply. It is confusing to have hidden preferences that conflict with the visible one.

Especially since directly below the visible one is a list of allowed sites which correspond to the hidden pref, not the visible one (or maybe it corresponds to both? I don't know?)