Is my circuit breaker box grounded? If not, can it be?
It's unlikely that your service panel ("circuit breaker / fuse box") is not grounded.  That could lead to all sorts of weird problems, not to mention that it would be tantamount to malpractice for an inspector and an electrician to walk away from an electric service in that condition.  Furthermore you said that some outlets are grounded.  If they passed inspection then you have grounded service.

(Technical clarification: usually the ground and neutral buses are connected in the main panel, and the service is grounded upstream.  By code such a connection can only be made in the main panel.  In this case the main panel is still "grounded" even though one might pedantically argue that the ground connection is not at the main panel.)

It sounds like you've gotten confused, so here's the simple situation: Most of the outlets inside your house are ungrounded.  They're the ones with only two holes, that can't accept a three-prong plug without an adaptor.  That's it, and it's very common in construction of that vintage or older.

The only way to provide grounded outlets is to run new wires from the service panel and replace the receptacles.  It's usually quite a pain/expense to do that, and apparently it's not required by your local ordinances.  (NB: See @wallyk's comment: You may have 3-conductor wiring to outlet boxes, in which case switching to grounded receptacles is a relatively easy project you can do yourself with the power off.)

One tip: If you do opt to pull new wires so you can ground outlets use 12-gauge wire, not 14-gauge.  That way you can support 20-amp circuits, which I have found comes in handy quite often, and the marginal cost shouldn't be that significant.