The Silmaril cast by Maedhros into a chasm, is in Middle-Earth, deep in Middle-earth in fact; and is one of the oldest objects in it, having been crafted by Fëanor in the early-to-middle First Age during the Valarin Years of the Trees, before the Third Age of the Unchaining of Melkor. Each Valarin Year was something like 144 years of the Sun so that is quite a long while ago. The One Ring in comparison is only slightly younger than Elrond.
Maglor's Silmaril while not technically in Middle-earth (it is at the deeps of the Great Sea) is more theoretically accessible. However, these objects are prophesied not to be reclaimed until the breaking of the world and Dagor Dagorath.
Andúril, along with Angrist, is said to be forged by the dwarf-lord Telchar "in the deeps of time". Presumably this means in Beleriand, in the dwarf-kingdom of Nogrod before the birth of the Sun. (I always thought Angrist was of Valarin make, but apparently not.)
Anglachel and Anguirel also date to this stars-only period, being made of star-metal by the Dark Elf Eöl, a student of Telchar. Anguirel and Angrist are presumed lost (the dagger that pierced the Iron Crown of Morgoth, and Anguirel would have been lost in the wreck of Gondolin), but Anglachel definitely still exists somewhere since Túrin is prophesied to wield it against Morgoth in Dagor Dagorath. I believe it is broken like Narsil, though.
Of the other named swords (e.g. Ringil, Orcrist, Glamdring and the dagger Sting) only the former is probably of Valarin make, and the latter three are known to have been forged in Gondolin during the Siege of Angband, possibly prior to the Nirnaeth Arnoediad, the Battle of Unnumbered Tears, making them a measly 3-400 years younger than the Sun, so that doesn't count.
The Ring of Barahir is a sign of the House of Finrod Felagund and is definitely a Valarin jewel, crafted by the Noldor (possibly Finrod himself using gems greated by the House of Fëanor) deep in the Years of the Trees, making it older than any of the aforementioned swords, and more accessible than the Silmarils. It may be the only Valarin artifact you can touch.
The Elfstone given Aragorn by Galadriel (replaced by the Evenstar in the film) may be a Noldorin jewel from Valinor, making it an artificial gemstone and very old, but reset by Galadriel.
The Palantíri (the few that remain) were given to Men of Númenor by the Elves of Tol Eressea as a gift in the early Second Age. This suggests they were already on Eressea, and the count of seven suggest a connection to the sons of Fëanor who allegedly made them, plus they are said to connect to a stone at Avalloné on Eressëa and possibly the master-stone at Tirion in Valinor, suggesting they were made during the Years of the Trees.
The Nauglamír (the Necklace of the Dwarves) was crafted by the Dwarves (Telchar?) as a gift for Finrod Felagund. Húrin recovered it from the ruins of Nargothrond and gave it to Thingol in Doriath. Thingol had a Silmaril set on the necklace, and later had a not-insignificant falling out with the Dwarves who did the work for him. But this was lost when the Silmaril became a star, unless it was reset before then. Either way it's not in Middle-earth.
Durin's Axe uncovered by Balin in Moria may refer to Durin I, who governed Moria from shortly after the birth of the Elves (YT 1050, thousands of years before the Sun) until his death in the First Age, when it was passed on to Durin VI (eaten by Balrog). So that is possibly one of the oldest, but we don't know when it was made, or if it was indeed the axe of Durin I.
Of these, I would say the Ring of Barahir is the oldest artifact for which we have provenance.
'But Arvedui did not take his counsel. He thanked [ the chief of Lossoth ], and at parting gave him his ring, saying: "This is a thing of worth beyond your reckoning. For its ancientry alone. It has no power, save the esteem in which those hold it who love my house. It will not help you, but if ever you are in need, my kin will ransom it with great store of all that you desire."
Return of the King Appendix A: "Annals of the Kings and Rulers" I: "The Númenórean Kings" (pace Jason Baker)
P.S. If Anglachel is buried with Túrin then it still exists under a standing stone on the island of Tol Morwen (roughly the Mid-Atlantic). I cast Detect Treasure!