Where is this island, and why is the title of this question "Giraffe Necked Island?"
Identifying an island so obscure and small (only about 30 miles long, judging by the scale key in the bottom-right of the image) without using Google Maps (since I figure that's what the no-computers tag is here for) is a pretty tricky (if not nearly impossible) task. However, I thought it might be an interesting exercise to attempt this using an old trusty friend: my 'Special Edition' Family World Atlas, published by Insight Atlas in 2009 (the last physical atlas I think I ever bought, and which still sits beside my bookshelf rather than on it, as a result of it being too tall... One reason the physical atlas has fallen out of favour with many...).
I was able to identify the island by shape, and fortuitously quickly thanks to one assumption that (luckily for me) turned out to be true...
I noted that the OP's region - shown in grey at the bottom of the image, to the right of the copyright notice - was set to 'United States'. (The fact that the OP is US-based could also have been corroborated by a quick check of their SE profile page, but I somehow neglected to do that until just now...) I then chose to assume that this island might have some kind of trivia-worthy feature of significance to somebody based in the US, and therefore (another big assumption) it may well be an island in North America. So I planned to look up these pages in my atlas first and foremost, in the hope that I might find something there.
I got very lucky in that an island that looked about the right shape (albeit pretty tiny) could be seen on one of the first pages in this section:
St. Matthew Island in the Bering Sea - a place that turns out to be notable for its remoteness, completely uninhabited by humans at present.
In fact, overleaf on the next page there was a more zoomed-in image in which it could be seen a little more clearly:
This definitely matched the shape of the island in question, although it had been oriented differently in my atlas - and it was this change of orientation that made me realise why the OP had chosen the title for this puzzle:
Feedback: I would say this question felt more like a 'treasure hunt' than a 'puzzle', per se. If the island in question had been much smaller I suspect it would have been almost impossible to identify computer-less without an awful lot of manual searching, which would likely be tedious rather than fun. I definitely got lucky with my starting point - it could have taken me a lot longer if I'd started looking in a different part of the world first! Still, it gave me an excuse to dig out my old atlas and spend some time with it, and for that I am grateful...