What others have mentioned without going into detail is how the system would work with an RCD and faults.
It will depend largely on whether the neutral point at the substation transformer is actually connected to earth or not, if it isn't then the RCD won't work (although you shouldn't be able to get a shock to earth in this situation)
Unless it's a very small system, then I would suggest that the combination of leakage, capacitance, and faults in various parts will mean that it's highly unlikely to behave as a true floating system would be expected to - i.e. even in the absence of faults it would be possible to get a shock.
But back to the OP.
If your appliance develops a l-E fault, then without any earth connection the casing will become live - it will continue to work normally (assuming the fault doesn't itself interfere with operation), but will be dangerous to touch. Clearly some of these appliances will be within touching distance of extraneous earths, such as you'd find with a metal water pipe that's buried in the ground. So if you are leaning on the sink when you touch the (faulty) washing machine then you'll get a shock from the full mains voltage - but it is unlikely that the current will be sufficient to trip any breakers/fuses.
If you have an RCD in the circuit, then the fault in the appliance will still not trip anything - it'll sit there with a live case, and still working. But when you touch it, at the same time you have a conductive path to earth, then the RCD will detect the imbalance (not everything going out the live wire will be coming back on the neutral wire) and trip - hopefully quickly enough to prevent the shock being fatal.
But what if you aren't touching a sink that's earthed via the pipes ? Well then the question is how well earthed are you ? There is always some linkage via capacitance, but if you are in dry shoes & socks, on a dry floor, you might not get a shock - just a bit of a tingle, and not enough current to trip the RCD.
I recall many years ago I worked on a farm (I also knew fairly little about electrics and safety, and had no test equipment) - this was before RCDs were common. We had a free-standing water heater, and started noticing a tingle when we touched it. We were always wearing rubber wellies, so even though we were stood on a wet floor which would provide a fairly decent earth, it did no more than give us a tingle. When I looked, the contacts for the earth pin in the socket (an old 15A round pin one) had been bent - so I squeezed them back together, and when I switched on, one of the elements went with a bang.
So yes, adding one or more RCDs will vastly improve your personal safety.
It could be just one in your supply - but it's generally advised that you split the installation across at least two. Having multiple RCDs, each protecting part of the system, means that there is less leakage current so nuisance trips are less likely - but also means that if one does trip, you still have some circuits working.
At a pinch, using a plug in RCD will give some protection - but only to faults with whatever is plugged into it.
When picking an RCD, be aware that some (notably many of the ones built into RCBOs - that's a combination fo RCD and the overload function of your existing breakers) have a white or cream wire coming out that needs to be earthed. Avoid these, there are plenty of RCDs that don't have this.
Also avoid any time-delayed versions - these won't trip fast enough to stop you getting a fatal shock. They are used (simplification) where it's required to protect part of the wiring system (e.g. a sub main), while still providing discrimination with non-delayed RCDs downstream. Without the delay, a fault on any downstream circuit would trip both the RCD on that circuit and the one further upstream.
For safety, it would be a good idea to add earthing to all circuits - though I realise that's going to be impractical for some (particularly rented places), and an upheaval where it is practical.
But the principle of the majority of electrical system in residential settings (it can be different in a commercial/industrial setting with trained staff monitoring things) is that all exposed conductive parts are earthed - so that if there's a fault, the current that will then flow will blow a fuse or trip a beaker, thus disconnecting the supply.
There is also the idea of double insulated appliances where there is extra insulation so that a fault between the live supply and any exposed metalwork is made very unlikely - but you won't find washing machines built to that standard.
And of course, whatever you or the others on the forums decide to do, you'll need to find out what the local rules/regulations are. For example, in many areas, only licensed electricians are allowed to do any electrical work.
