You mean the island of Fyn in Denmark? You know, I checked online, and the only hit that claimed to have any knowledge of the "elves" was "realfairies.com", which I hardly think counts as a source properly based on empirical results.
I will try to be neutral, as although I find the whole idea ridiculous, I have had some mythologism as a part of my History classes at university, so I have spme eperience with tales of the supernatural and people who believes in it.
Now, the following is merely my own memory from lectures on Celtic and Norse mythology, so I reserve the possibility for minor mistakes.
What the "Elves of Fyn" are claimed to be has the Celtic version of elves in nature, a version that for example manifests itself in the Leprechauns (although the site claims them to be something else, which I will come to soon) of Ireland. However, their appearance appear to be based on the Norse elf version, made famous through Tolkien. The Norse human-sized elf does in Norse mythology live in another world, Álfheimr. However, this place is in no way similar to the human realm, Midgard, but rather covered in forest. The light elves, all blonde, are "fairer than the sun to look at". The dark elves are all black-haired and live undergound in another world. However, elves have not existed in Scandinavian mythology for more than a millennia. The only creature remotely similar to elves are the "nisser", small gnome-like creatures looking like Santa that kills livestock unless you sacrifice porridge to them on Christmas eve, a remnant of the Norse winter sacrifice to the gods, taking place the 22nd. If the elves were real, the myths would have remained, right?
Anyway, there is no empirical or scientifical evidence for the existence of elves whatsoever. Several times in history people have claimed to, as with all kinds of mythological creatures, prove their existence. However, it has all been exposed as fraud over time, people trying to become rich or famous. I can with 100% certainty tell you that elves do not exist, and certainly not living in a world corresponding to ours. And even in Norse mythology it was not possible to cross between the elven world and our world. Only the gods could pass in and out of their world, or lead elves out, as the lack of a Bifrost connection necessitated travelling along the invisible roots of Yggdrasil, the locations of which are known only by the gods.