The immortals are among us. And they have been for a long time. All of them were born in the 7th to 4th century B.C. in the area of what today is the regions of Macedonia and Thessaly in Greece. Whatever caused their immortality is no longer present. As far as they know no other people with this affliction exist.
They are not truly immortal. They don't visibly age and always appear to be in their early twenties, but they can be killed as easily as normal people. Violence, accident, drowning, starvation or a severe disease can kill them, but if they can survive the first 24 hours they will usually make a full recovery.
They heal about 25% faster than a normal human. Their immune system will eventually deal with any disease or poison and they will regenerate any damaged tissue (even nerves or brain-tissue) completely given enough time. It will take about 20 years to regrown a lost arm or leg.
It isn't all good though. They are all infertile.
In ancient Greece being immortal wasn't really a big deal. People simply assumed you were one of Zeus (or any of the other gods) many illegitimate children. Demi-gods are supposed to have strange powers.
But in later centuries this changed. The immortals quickly came to realize they better stay in hiding. Especially the Christian church took a dim view of people with special powers, proclaiming them to be witches or demons. A lot of immortals ended burned at the stake.
So the immortals went completely underground. They moved from town to town every 15 years or so to prevent the neighbors noticing the everlasting youth. Cosmetics to appear a bit older have helped too. Cosmetic surgery is no good though. Their regenerative system starts to repair the damage straight away, undoing the changes.
It was especially difficult for the immortal women as they couldn't easily operate solo for most of our history. They would usually team up with one of the immortal men and pretend to be a married couple (and in some cases they actually were a couple).
Of the original about 4000 immortals some 800 are left in 1950. They are mostly based in Europe and the America's. They have a central organization masquerading as a legit international trading and shipping company established in the early 18th century, that keeps in touch with them. Another 200 to 300 of them have been out of contact for decades, but they may still be around somewhere.
After World War II, in the 1950's they start to realize that identity papers and civil records are becoming ever more important in any modern society and assuming a new identity every so often is becoming increasingly difficult every year. Later on computer records and bio-metric ID's made things even worse.
So how can they keep hiding from 1950 onward to today?
Preferably while leading a fairly normal life with all the amenities? Living completely off-grid somewhere in Alaska, Siberia or Africa doesn't appeal to most of them.
Money is of no concern. The organization has been around for centuries and has invested wisely. They are not adverse to illegal activities if those are necessary, like bribing people, tax evasion, forging documents, but they draw the line at violence. No murder of inconvenient witnesses.