Evermeet is currently on the Material Plane, or at least reachable by ship
Current lore sources from 5e strongly suggest that it has a physical location on the Material Plane.
Map of the Sword Coast
Perhaps the most authoratiative source is the 5e Wizards of the Coast published Map of the Sword Coast which clearly has an arrow to the west of the Moonshae Isles, pointing due west, with the label "West to Evermeet".
Sword Coast Adventurer's Guide
Aside from that we have various passages from the Sword Coast Adventurer's Guide which are presented as authoritative:
Chapter 1: Welcome to the Realms
This chapter is presented as a factual set of information about the Realms and it's people.
Chapter 1: Welcome to the Realms > The Sword Coast and the North > Island Kingdoms:
Off the western coast of Faerun are a number of island relams varying in size. The most distant, and yet perhaps the most symbolically important to the mainland, is Evermeet, the island paradise of the elves, reputed to be a part of the divine realm of Arvandor. [...]
Chapter 1: Welcome to the Realms > Toril and it's Lands > Beyond the Trackless Sea:
Farther to the west, past even Evermeet, are untold, unknown lands beyond the Trackless Sea.
Chapter 2: The Sword Coast and the North
This chapter is intentionally presented as a set of first hand accounts. The opening paragraph even warns us that what follows may not be reliable. It states:
This chapter details many of the locales of the Sword Coast and the North, as seen through the eyes and recollections of a person living in Faerun. Rather than being exhaustive descriptions, what follows are snippets of information drawn from the experience of five individuals who have traversed, lived in, and explored these areas. Like any other narrators, they have opinions and biases, and may be drawing conclusions from incomplete information. No one in the Realms knows everything about any subject, even its oldest and most learned sages, and the views formed from such incomplete information can often suggest an inaccurate conclusion. This is not to say that any of the information the narrators provide is false, only that they may not be entirely knowledgeable in their declarations.
As a result, we should take what is presented here with a grain of salt (compared to the more authoritative information presented in Chapter 1, which doesn't come with any such caveats).
All that being taken into account, there is some relevant passages to our investigations presented here.
Chapter 2: The Sword Coast and the North > Island Kingdoms > Evermeet > A Little Piece of Heaven:
Legend has it [...]
Then the Spellplague struck, and some of that old elven High Magic must have unraveled. Evermeet became unmoored from the world and found itself instead in a sea of the Feywild [...]. For a century, it seemed Evermeet was lost to the world. [...]
Their patience was [...] at last rewarded, when ship from Evermeet docked once more in Sword Coast ports.
This passage, suggests that during the Spellplague Evermeet was disconnected from the Material Plane and stuck in the Feywild for a century. Since the end of the Spellplague it suggests that the island is once more moored to the Material Plane, as evidenced by the ships from it arriving in various ports of the Sword Coast.
Chapter 2: The Sword Coast and the North > Island Kingdoms > Evermeet > Sailing to the West:
I overheard it said by the captain of the ship that Evermeet somehow straddles all three planes: our world, the Feywild, and Arvandor. It touches them all, but exists fully in none of them. To find it, you must follow a pattern of stars until the stars change and then follow new stars. (I swear by Garl's mugget that's what he said!) [Those who stray from the path are lost. How I wish I could have asked the captain where the lost ships went! But I couldn't give myself away.
This is obviously presented as someone remembering something oveheard illicitly, said by someone else (the captain) to an unnamed third party as part of a conversation.
It does however give us two pieces of information (of various levels of reliability):
- Evermeet can be traveled to by ship on the Material Plane (the speaker claims to have done so)
- Evermeet is potentially present in three planes at once, but fully part of none
Chapter 3: Races of the Realms
This chapter returns to a more authoritative style, and doesn't rely on the flourishes of third hand information that chapter 2 presented.
Chapter 3: Races of the Realms > Elves:
[...] the days of the great elven nations are now long past, and many elves have withdrawn from the world into isolated sylvan realms, or set sail across the Trackless Sea to the isle of Evermeet.
This clearly presents Evermeet as an island located "across the Trackless Sea".
It also states:
At the height of their power, the elves performed a High Magic ritual intended to create the ideal homeland. They succeeded, but the spell sundered the land in a terrible cataclysm at the same time that caused the distant isle of Evermeet to rise from beneath the sea.
This sets Evermeet as an island created by great magical forces, which may lend credence to the third-hand, overheard information that Evermeet is part of but not apart of three separate realms at once.
Chapter 5: Backgrounds
Finally, as part of one of the new/augmented character options presented in this book we have the Far Traveler background.
Chapter 5: Backgrounds > Far Traveller > Where are you from?:
Evermeet The fabled islands far to the west are home to elves who have never been to Faerun.
This snippet further establishes Evermeet as physical islands "far to the west" of Faerun.
Mordenkainen's Tome of Foes
The last bits of lore we have in 5th edition come from Mordenkainen's Tome of Foes. This entire book is presented (in the Preface) as being:
the true work of that Oeridian wizard Mordenkainen, although penned in large part by his apprentice, Bigby. [...]
Bigby wrote as dictated by his master, burdened by the chains of a charm spell. Eventually Mordenkainen released Bigby from the spell once he had turned Bigby away from his evil ways and Bigby had earnede Modenkainen's trust. The last few chapters are thus penned in Mordenkainen's own hand. But you'll see the voice of the author is the same throughout.
Later in the Preface it states
A companion to the Monster Manual and Volo's Guide to Monsters, this book contains the musings of the renowned wizard Mordenkainen from the world of Greyhawk. In his travels to other worlds and other planes of existence, he has made many friends, and has risked his life an equal number of times, to amass the knowledge contained herein.
With that introduction we should bear in mind the warning presented in Chapter 2 of SCAG:
No one in the Realms knows everything about any subject, even its oldest and most learned sages, and the views formed from such incomplete information can often suggest an inaccurate conclusion.
... and thus take the information presented here, as probably true, but not necessarily authoritative or entirely accurate. We also don't have information on precisely when the information in it was gathered by Mordenkainen, so it's possible that all of the information in the book was accurate when it was gathered, but is now rendered inaccurate or out of date due to changes in circumstance. This is particularly important when we come to the section concerning Evermeet.
With that in mind, Mordenkainen's Tome of Foes does have one section on Evermeet of potential interest to us.
Chapter 2: Elves > The Seldarine > Evermeet:
[...] By means of a cataclysmic ritual, the greatest elf wizards of Faerun summoned into the world a piece of Arvandor and bound it there.
[...] Milennia later, Evermeet still exists, although now it is unmoored from the world, somewhere in the space between the Feywild, Arvandor and the Material Plane. By using secret pathways, entering a fairy ring on special nights, or traversing a moonlit sea by following certain stars, elves of many worlds can get to Evermeet - if they're lucky. Even from Faerun, for instance, one can sail to Evermeet only on a ship captained by an elf who has been there before. And if the captain slips up, the ship might become adrift on the Astral Plane.
This suggests, in conflict with our other sources, that Evermeet was once tethered directly to Faerun, but is now unmoored (it's interesting the same language is used as the third hand account from SCAG) from the Material Plane, but can still be sailed to it from the Material Plane.
Conclusion
Taking all of the above into account, the most authoritative 5e sources we have strongly put forth that Evermeet is a physical island on the Material Plane, but that it may be, at the same time, between three separate planes.
5e doesn't give any specific distances to Evermeet, beyond that it's at least 1,000 miles from the Sword Coast, as that's where the map ends. Sources from other editions can give us more lore, and I will expand this answer with that lore when I have time.
But for now, we can estimate that the journey is somewhere between 1,000 and 3,000 miles away from the Sword Coast (just to put an upper limit on it). The distance in our world, from Europe to the Americas is ~3,000 miles (eg Ireland to New York). According to Spartacus Educational:
In the early 19th century sailing ships took about six weeks to cross the Atlantic. With adverse winds or bad weather the journey could take as long as fourteen weeks.
So 6 - 14 weeks for a 3,000 mile journey, which would translate to 2 - 4.5 weeks for a 1,000 mile journey.
Using our estimate, that means sailing to Evermeet from the Sword Coast could take anywhere from 2 - 14 weeks of sailing, depending on the weather and the actual distance from the port of origin.
