Where do the people go when they leave Middle Earth?

Answer 1

They go to the original land of the Elves. Where no one can die. It is like "Heaven on Earth" in some ways. In LOTR, the chapter when they leave is called "Grey Haven" so that might be what it's called.

The Gray Haven is a port in Middle Earth that the boats that take the Elves away leave from. The place they go to is called Valinor, and by the end of the War of the Ring it is no longer physically located on Earth at all. It is also not the 'original land' of the Elves, since they are from Middle Earth. However the majority of the Elves travelled to Valinor early in their history, and the ones we meet in the books are the ones that came back.

Answer 2

In the Silmarillion and his other writings, Tolkien describes a cosmology where there is a supreme deity (Eru) and lesser beings (the Ainur). Some of the Ainur (called the Valar and Maiar) came down to live and shape Middle Earth, ultimately settling in the extreme westernmost continent of Valinor. At the end of the Second Age of the Sun, Valinor was detached from the rest of Middle Earth by the power of Eru and is now accessible only by the ships of the elves.

Many elves live in Valinor and much of the plot of the Silmarillion revolves around events that lead to some of these elves choosing exile. It is decreed that these elves may return to Valinor someday on the grey ships. At the end of LOTR, Frodo is permitted to journey with them to Valinor.

Valinor is thus an actual physical continent, to be found in the extreme west or later floating near the world. But Valinor is also in many senses heaven. It is a place where angelic beings dwell in beautiful surroundings and death, disease, and suffering are virtually unknown.

Answer 3

the undying lands, where the elves live, beyond this reality... heaven?