In the Akallabeth, the Elves or the Valar say to the Numenoreans that it is not the land that makes the people holy, it is the people that make the lands holy. It was the Elves who were immortal, not the land that made them so. The Numenoreans thought that if you were a mortal living in Aman, you would automatically become immortal (leading to one of the stupidest acts in Middle-earth's history).
I don't think that Frodo, Sam, Bilbo, and Gimli would have become immortal. Only Eru could have granted that, and I don't think he would have chosen to. They probably all passed away peacefully, with all their friends beside them.
After death, they probably went to the Timeless Halls where all the other spirits of Dead Men awaited. They probably met Merry, Pippin, Aragorn, and Boromir again there.
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But Melkor also was there, and he came to the house of Fëanor, and there he slew Finwë King of the Noldor before his doors, and spilled the first blood in the Blessed Realm; for Finwë alone had not fled from the horror of the Dark.
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