Yes, yes, の is like 's, but I’m saying that 現実の (the whole block) acts like an adjective. There’s actually something called a の-adjective , but if that sounds confusing (I certainly don’t use the concept much), you can see it as の linking a characteristic feature to the thing being characterised. (I think @YanagiPablo mentioned this idea before.) AのB can also be ‘B characterised by A’, just like how ‘of’ in English and “de” in French sometimes link characteristic things to the ‘main’ noun, like ‘a man of steel’ = a man made of steel (literally)/who is like steel (figuratively). I’m fairly sure we’ve all come across 普通の as in 普通の建物. 普通=the ordinary, so 普通の建物 = buildings of the ordinary = ordinary buildings.
In your sentence, if we use my 's interpretation, we get 現実の日本の医療費=reality’s Japan’s medical fees=medical fees of Japan of reality, but the ‘reality’ bit doesn’t make sense if we translate it like that. However, if we take it that ‘reality’ characterises ‘Japan’s medical fees’ in some way, we might say ‘real medical fees in Japan’ or ‘actual medical fees in Japan’.
@YanagiPablo: 崩壊 is also the name of an anime-style game produced by a Chinese developer which has Japanese voice actors to do the Japanese dub! (The official English name is more of a compromise between the Japanese and Chinese pronunciations of 崩壊 though…) More seriously speaking, in Chinese, 崩=collapse/fall apart/burst and 壊=broken/bad (this meaning doesn’t exist in Japanese, but in Chinese, it’s the everyday word for ‘bad’; 悪 is a literary word for ‘bad’, and is sometimes closer to ‘evil’ in Chinese).
By the way, just a general thought for everyone about the particle で: sometimes, I just treat it as the mid-sentence form of だ・である・です when the ‘means’ interpretation doesn’t really work. I mean, で probably gave its meaning to である, and not the other way around, but that doesn’t mean we can’t apply our understanding of である to で. E.g. このままでは: the condition/state of the medical system isn’t really a ‘means’ to cause its finances to collapse; it’s the cause of the collapse. No one’s using it to bring about the collapse. It’s more like ‘if this state continues’. On the other hand, if we treat it as a form of である, we get ‘This state being (the case)…’, and the sentence makes sense.