In years gone by piston engined aircraft had unreliable engines, which is part of the reason why long haul aircraft had 4 engines. It was quite usual to see an aircraft with one propellor stopped (not frequent but unremarkable). Rules were crafted to protect people from making errors and one was that if you only had two engines you had stay within 60 minutes of a diversion airfield at single engine speed. That's a gross simplification of history.
Those rules were still in operation when airlines (I think TWA was first) started to think about operating 2 engined jets across the Atlantic. In fact TWA had to stick by those rules for the first flights they made with a 767 across the Atlantic. But everybody recognised that the latest jet engines were much MUCH more reliable than the old piston/propellor engines. After much debate a new set of rules were crafted covering the operation of twin engined jets across oceans. This is ETOPS which Tightslot has already referenced for you.
I can remember being asked to visit TWA on business not long after they had started operating St Louis/London using a 767 and even though I have always enjoyed (and still enjoy) flying when the suggestion of using the route via St Louis was suggested I balked and we went via Chicago on a 747.
Then an acquaintance who was a pilot started flying the 767 on long haul routes. He explained the new ETOPs rules to me and I begun to appreciate that they had been crafted in such a way that it was arguably safer to fly on a twin jet over water than a quad. Surprisingly (it may have changed) he could depart on a flight in a quad knowing his destination airfield was open but that all the possible alternates en route were predicted to be closed. In a twin the alternates had to be available as well as just in existence. And there were other things that had to be on a twin like extra fire protection that didn't have to be on a quad. So, I started to fly on twins as well as quads and now I don't even think about it.
What's interesting is that, slowly, all those little extra requirements that were imposed on twins have slowly made their way on to quads, enhancing and improving the safety of all planes.
So, what happens if an engine stops en route? You divert to somewhere that can fix it. How often does it happen? Rarely; very rarely. What happens if all the engines fail? Well, that's even more rare (although if you've been reading this site you'll know it has happened recently) and, in many cases, crews will restart engines and then you can divert. If they all fail and stay stopped, it doesn't matter whether it's a quad or a twin you glide until you have to land.
Medical emergency? Depends on the nature. Almost all airlines have radio access to a doctor for advice and, if necessary, the aircraft will divert to somewhere with appropriate medical facilities (not always the nearest airport).
And, that's something to remember. When an "emergency" happens on a flight, it's not always appropriate or necessary to divert in a mad dash to the nearest airport. There's little point in landing at an airport where there is no hospital nearby if the emergency is medical. And, even if there is a hospital, it may not have the required facilities so you need to go further to find a hospital to maximise the chances of the patient. And that can be as true over land as over water; the "correct" airfield may not be the nearest.
In the end I always take heart from the fact that the crews on planes are people, just like you and me. They don't board a plane in the expectation of something going wrong. Just like you they want a hazard free journey and they will do everything they can to make sure they (and you) arrive safely. They are, however, trained in how to react should the unexpected happen; trained extremely well.