The Grey Havens. What exactly is it?
I've always been fuzzy on this and whenever I ask anyone, I get very vague answers and even more vague hand gestures. Am I wrong in that the bare bones idea of it is that it definitely exists on a map, and elves and very select non-elvish people go to spend a physical eternity there when they feel they just can't hack Middle Earth anymore? Sure, Bilbo and Frodo got to go, but did they live forever over there? And it seems to be understood that you can't just get in a boat and go check it out to see what it looks like. Is that really all there is to know about how the Grey Havens works?
The other short answer is that if someone who's not an elf goes to "The West", then no, they do not become immortal. I'm told that, in fact, your life gets shorter, though it's a bit like the greatest hospice ever. You can heal and rest up, but if you stay there the paradox of being a non-elf in eternal elf land will kill you. But you can come back. I say "The West" because, oh man, does it get nitpicky where Frodo actually went.
For the thorough answer, I cite Mr. Paul Inglis:
They're about 100 miles to the west of the western borders of the Shire (approximately - I'm not looking at a map).
They are certainly a real place on a map - but I think that maybe you are confusing the Havens, the departure point of the Elvish ships, with their destination in the Undying Lands (not enough of a Tolkien geek to be able to name the exact place, but it's somewhere in Eressea). The harbour of Alqualonde springs to mind. This stuff is all from Tolkien's earliest imaginings, about 25 years before he started writing 'The Lord Of The Rings'.
You won't find the destination on a map in 'The Lord Of The Rings', nor can you get there, unless you're an Elf (or have very good Elvish connections). The Undying Lands are in a sense a kind of Earthly Paradise, and they were *originally* part of the real physical world - but they were eventually "removed" from the Earth in terms of being accessed by regular folks, at least. However, in some sense they are still physically "there" - but not for the likes of you and I - unless you are especially favoured. If you were able to travel there you would feel the sensation of your boat being guided onto a "Straight Road" in the ocean that would seem to lift away from the curve of the Earth, while passing through mists. After a time there might be a warm rain and a feeling that perhaps you have passed through a curtain of silver, and then you would in time arrive at lamplight quays under a tall mountain, and perhaps see "white shores and beyond them a fair green country under
a swift sunrise". But I've said too much already.
Tolkien got the idea (which as I say is part of his very earliest writings) from a handful of Anglo-Saxon poems that deal with seafarers that came to a land across the western ocean from Britain - presumably America we moderns would surely think - but what they found was a land of enchantment and wonder inhabited by people described as "Elves". However, when later seafarers tried to retrace these voyages they (at least the ones who returned) found only regular lands inhabited by humans.
Curiously, the Elvish harbours in the Undying Lands were said to lie "nigh to the girdle of the World", the Equator. In other words, quite close (presumably) to the Caribbean and (for that matter) the Bermuda Triangle (not that Tolkien said that, I'm just noting their position relative to places in our world).
As for the Grey Havens themselves, they are an Elvish settlement on the western coast of Middle Earth (think north western Europe) to the west of the Shire. There were orginally other Elvish Havens from where Elves might return to the Undying Lands (for example Edhellond in Gondor) but they were abandoned as the number of Elves steadily diminished. You can certainly return from the Havens, as Sam, Merry and Pippin do.