I’ve got plenty of fossils, many of which I found myself, but I really don’t think that counts.
I’ve got a copy of a classical Greek book from the 18th century. I also have what I believe is a very torn print by Hogarth. It’s precisely the same size as the original, and has clearly been “pressed” from a copper master. That’s be circa 1725.
A have a sackful of Roman coins I ordered from a catalog. They arrived unsorted, unidentified and still caked with dirt – supposedly the only thing known about them was that they were bronze. I cleaned them off as best as I could and consulted a few library books to try and identify them. Most were little better than slugs, but one of them I actually managed to peg. I don’t recall the details of it, and since it was stolen I’ll never be able to trace it again, but the condition of the coin was certainly nothing that would suggest any great value.
I have two other Roman coins. One that my father ordered for me as a gift. It’s a silver denarius of Septimus Severus, in fairly good condition. I’ve never taken it out of the plastic pouch in which it arrived, largely because it could very easily be mistaken for a dime and disappear into a vending machine.
The other has a strange historical twist to it, and I keep it in my wallet as a conversation piece. It’s a bronze as from about 317 AD, during the reign of Constantine (like this but not in as fine a condition). The obverse shows the great man himself, Constantine I. The reverse features Sol Invictus, the god of the “Unconquered Sun”. In other words, on one side we have the first “Christian” emperor, and on the other, the pagan god whose feast day was celebrated at the Winter Solstice – December 25 by the Roman calendar. When I ordered this coin from a catalog, by the way, I knew only that it would be a genuine bronze coin, and had no indication there would be anything of significance to it.
A couple of Jane Austen books from about the late 1800s (I’ve never really had them examined or appraised or anything, although I want to very badly).
-Various shards and pieces of painted and unpainted Greek Pottery (Age unknown, but likely thousands of years old.)
-An emigration Document for my Step Great Granddad from 1871, Issued by Das Königsreich Baden-Wurtemburg (He was from Rottweil.)
Manmade would be some Archaic Period projectile points found in the woods where I grew up and probably around 3,000 to 5,000 years old.
Oldest family heirloom is a churn that’s been in the family for about 170 years.
Oldest think altogether is probably a meteorite (about the size of a brick but weighing slightly more than Cuba except for when it glows, when it’s light as a feather but hums).
As far as manmade objects, I have a Southern Cross of Honor medal which was given to my great great (great?) grandfather after he fought in the Civil war.
In the natural objects category, I have rocks with fossils of seashells, found on the top of West Virginia mountains. Those have to be pretty damn old!
A journal I kept when I was 17. I’ve considered destroying it. When I read the stuff i wrote in it I’m embarrased for myself. Gawd, I was a little putz…I can barely believe I wrote that stuff.
Well, no…older than that I have an photo of my mother when she was in High School wich was in the 1940s.
I used to collect Roman and Georgian jewelry, so I have a few Roman key rings and other stuff and some Georgian mourning jewelry.
After that some family stuff from the eighteenth century and on down.
Sometimes I think I’d actually like to have something brand NEW in the house! The only big thing I use on a regular basis that somebody didn’t own before me is a kitchen table I got at Ikea. 
A vial of residual matter from the Big Bang… 
that, and the family heirloom Parker VH grade 12-gauge side-by side shotgun built in the 1920’s and has been in the family for three generations
A full set of Dickens from the late 1860’s is, I believe, my oldest.
But, my copies of A Midsummer Night’s Dream and Romeo and Juliet from 1880 are my pride and joy (A Midsummer Night’s Dream is my favorite Shakespeare).
(and just to brag)
While certainly not in the top 20 of the oldest things I own (all books, BTW), one of my most beloved possessions is my first US edition of Carry On, Jeeves from 1927.
I just remembered our bedside table. It’s not as old as the photographs I mentioned, but it was made in the 1920s. Teakwood, and came from Burma.
Maternal grandmother’s encyclopedia, dated 1923, with quarterly updates through 1931. (And, now that I look at them, several postage stamps still affixed to the envelopes.) Grandma was so respectful of books – but so insistent on truth – that she made a correction in pencil when her town was placed in the wrong county.
Same grandmother’s dining room table and buffet which were delivered to her new home the day before she married, in 1924.
Paternal great-grandfather’s pocket watch, which I’ll give to my grandson when he graduates college.
Paternal great-grandmother’s footed cake plate, which came across the Atlantic when that family immigrated in 1902.
A great aunt’s Seth Thomas mantle clock: still runs, still chimes, and I have no idea how old it is.
There’s probably more… my sis and I are the only grandchildren on both sides so stuff has filtered down to us.
Besides fossils, the oldest is an 1865 English penny, in fairly poor shape. The oldest thing that is personal to me is the string of hospital beads with my name on it. It’s from 1960.
My home seems to be the repository for small family heirlooms. The current list:
*pictures, beginning with daguerreotypes from the late 1860's
the family bible given to one g-grandmother at her wedding in 1905
another g-grandmother's wedding band
my g-grandfather's pocket watch and Winchester hex barrel shotgun from the late 1800s
my grandfather's set of about a dozen literary classics from the early 1900s
a book case that the same grandfather made in shop class in 1916
arrowheads found by my g-grandfather, grandfather, dad and uncles while plowing*
The oldest item that has been with me since it’s existence is the quilt made for me as an infant by my g-grandmother in 1963.
I have some rocks and Civil War memorabilia I can only assume are genuine, but I definitely don’t know for sure and I wouldn’t vouch for what they sell kids in some of those tourist traps. So putting those aside, I’d say the two Mercury Theater Shakespeare scripts my grandmother gave me. I have those with me at my apartment, while the other stuff I mentioned is at my family’s house.
Personal possessions I actually use frequently? My pocketknife is about 37 years old. My .38 snubbie is about 27. I have books and stuff older than that, but these are things I carry with me daily.
My home is a 1868 Greek revival
(this probably won’t impress many outside the U.S. but for the Chicagoland area it’s pretty old - further, and much to my chagrin, it has had few rennovations, for example; we just replaced seven layers of shingles off of the roof! And it might as well be made of swiss cheese for all of its insulating ability…)
A few books on the American Civil War from the 1870s that are in good condition; one book from 1894 on the “Great White City” (Chicago’s World Fair 1893) that is like new (apparently I had some foresighted ancestors).
I also have dozens of fossils, mostly from the Eocene (@55mya) and a few much older (but I agree that these are like cheating as they are so resilient)
I have my mom’s father’s pocket watch, from 1915. He won it for having the best crop that year or something; it is inscribed on the back and was given as a prize to him in some sort of agricultural club.
It has real gold on the back and until recently it was still ticking. It has a long chain too.
I have to confess that every time I look at it, I’m reminded of the scene in Pulp Fiction—you know, the one with Christopher Walken. “This watch…”
The oldest item that I know of, is a $10 US Gold coin minted in 1907. It’s in pretty good condition too.
I have a couple of physics textbooks from 1904. One of them mentions Michelson’s work in the briefest passing; the other contains no hint whatsoever of any of what’s now called “modern physics”. I’ll bet those authors were surprised, the next year.