My old house has no ground wire — is fitting three-prong GFCI outlets actually safe?

What you do not get is worth listing precisely, because this is where people over-estimate the substitution.

• No path for fault current. With a proper ground, a live conductor touching a metal appliance case dumps current to ground and trips the breaker immediately. Without one, the case simply sits energised until somebody touches it — at which point the GFCI saves them, but the fault was live until then.
• No surge path. Surge protectors and many filters rely on a ground to divert energy. A surge strip plugged into an ungrounded outlet provides much less than its label suggests.
• No reference for sensitive equipment. Some electronics and audio equipment behave badly without one.
• Not everything is covered downstream. Protection extends to outlets fed from the protected one, and only if wired correctly.

So: good protection for people, absent protection for equipment. That is the honest summary.

The device does not need a ground wire to work, and understanding why makes the rest obvious.

A ground fault interrupter compares the current flowing out on one conductor with the current returning on the other. In normal operation they match. If some current is escaping by another route — through a person, through a damp wall — the two no longer balance, and the device cuts the supply within milliseconds at a very small imbalance.

Notice that nothing in that description involves a ground wire. It is measuring a difference between two conductors it already has.

So for protection of people against shock, which is the thing that hurts you, this genuinely works without a ground. Many electrical codes explicitly permit exactly this replacement in ungrounded systems, on the condition that the outlets are labelled to say there is no equipment ground. Those labels usually come in the box, and fitting them is part of the job rather than an optional extra.

From having done this in a house of the same vintage, two practical notes.

Do not connect the ground terminal to the neutral to make a tester show a ground. This is occasionally suggested and it is genuinely dangerous — it puts normal return current on the appliance case and makes an open neutral lethal. Any tester showing a proper ground where none exists is worse than one showing the truth.

A plug-in tester will report a fault on these outlets and that is expected. Those testers detect the missing ground and cannot detect a working interrupter without a ground. Use the test button on the device itself, monthly.

Also worth knowing that in some cases running a single ground conductor back to the panel for one circuit is permitted and is far less work than rewiring. Worth asking about for the circuits that feed your electronics.

One thing to raise with whoever checks the work: the condition of the wiring itself.

Houses old enough to have no ground often also have insulation that has become brittle, and connections that have been worked on repeatedly over decades. An interrupter protects against a fault to earth; it does nothing about deteriorating insulation between conductors or a loose connection heating up in a box.

So the outlet upgrade is a real improvement and it is not an assessment of the system behind it. Worth having somebody look at a couple of boxes while they are there.