Electrical – Is the rental house safe even if the electrical is not grounded at all

It is extremely unlikely that all three receptacles all have bad grounds. Not impossible, but unlikely.

The most likely scenario is that your home was built before grounding became required by code (1962, plus however long it took for your state/municipality to adopt it), and then someone renovating after that time put in 14/2 Romex with a bare ground for the outlets, but then found that the supply wire from the panel didn't have a ground and didn't bother to replace that wire. So, the ground is either shorted to the neutral or left open.

The other likely scenario is that there is a discontinuity in the ground between these three outlets and the supply wire. The necessary wires exist, but whoever last messed with your home's electrical didn't connect them properly.

An outlet tester, and a more thorough inspection of the wiring, can tell you. Head to Home Depot, Lowe's, Ace, whatever, and pick up a 3-prong GFCI outlet tester:

Plug it into all of the outlets on each receptacle; ideally, you should see the two amber lamps light, but that's unlikely if the surge protector doesn't detect a proper ground path. What's more likely is that you'll see only the middle lamp light, indicating an "open ground".

Unfortunately, because the neutral and ground are supposed to be continuous (they both tie directly to the neutral bus strip in the service panel), there isn't a really good way to detect a neutral-to-ground short or swap without opening up all the outlet boxes; a 3-lamp tester will read "correct", and a multimeter will read continuity between neutral and ground no matter whether they're correct, swapped or shorted. However, this scenario is very unlikely in your case, as the surge protector should read "grounded" if the ground pin has continuity.

With the symptoms confirmed (hopefully), turn off the breaker (or pull the fuse) for this circuit, and open up all the outlet boxes. Verify that all of the wire bundles coming into each of the outlet boxes have a bare ground wire, which is twisted together with any other bare wires in the box, and is also connected to the outlet's grounding terminal (it is also acceptable, if the outlet box is metal, for the wires to be screwed into the box itself; the outlet will then be grounded through its chassis and the mounting screws. If the box is a resin or thermoplastic, the ground screw of the outlet chassis must be used).

If that doesn't indicate the cause of the problem, you will need to determine which of the various wire bundles is fed directly fro the panel. This is generally done by disconnecting all the black wires from all the outlets, then turning the power back on and VERY CAREFULLY probing each one with a non-contact voltage tester, like this:

The black wire that lights up is the one fed most directly from the panel; however, it may not be the actual wire run from the panel itself, as there could be something further "upstream". The usual culprit is the switch for the lights, especially in older homes (newer NEC code encourages lights and outlets to be on different circuits, by encouraging calculation of demand load based on separate lighting and appliance branch components, so you're not fumbling around in the dark if your hair dryer shorts out). If your NCVT lights up on anything else that you know is "dead" when the power to this circuit is off, turn the circuit back off and pull those things out of their boxes to inspect those wires. Check for the existence of ground wires, whether they're properly continuous with all other grounds, and with all black wires disconnected, whether any black wire still lights up with the power on.

If the trunk that seems to be the main power feed does have a ground, use a multimeter to measure the ohms between the neutral (white) and bare (ground). It should be zero. If it isn't, the ground is probably not properly connected to the neutral bus strip at the panel; this is where you call in a qualified electrician, because the rails in a fusebox or breaker panel can kill you instantly with one wrong move. If the ground doesn't exist in that supply wire, that's the problem; you'll need to run a new length of grounded Romex to the panel, or alternately run a green-jacketed ground wire to another box with a known good ground (I THINK this is acceptable; an electrician can verify). Understand that if you have a fusebox and the electrician has to open it up, he may insist on replacing it with a breaker panel in order to bring everything he's touching up to current code.

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.