What is the oldest object that you own?
Mine is a 1799 “Draped Bust” silver dollar.
I have a child’s rocking chair made to commemorate the sinking of USS Maine. It’s pretty cool, and has been in my family since it was new. (shortly after 1898 I would presume)
Other than that, I’m not sure. I have a box of old stamps and letters that are older. Not sure where it is…
I have some old rocks and some arrow heads. They’re probably pretty old.
A fossilized trilobite from over 250 million years ago.
I have an 1845 five cent coin.
It is going to be hard to beat this, therefore, to hijack the thread slightly, which of your possessions have you had the longest? I still have some of my toys from 65 years ago.
I have an orange sad-faced basset hound (piggy) bank which has been with me for 25 years.
I will NEVER give him up, and he has saved from being broke many a time.
I’ll include a picture as soon as I can.
I bet some of y’all will recognize him and have one of your own.
He’s as precious to me as Bert is!
Quasi
I have a 1966 Cadillac Hearse, a 1962 South Bend Lathe, a 1952 Grundig Majestic Radio, a 1944 ‘Twas the night before Christmas’, a WW1 compass of unknown provenance.
That’s what comes immediately to mind…and it makes me feel old.
Winchester Model 1892 Rifle in .32-20, built in 1897.
(I’m not counting the trilobite fossil.)
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I’d recommend the spirit of the OP doesn’t include items of ‘geologic time’…heck, I can go to my back year and pick up a bajillion year old piece of quartz.
I have a doll with a china head which was brought from England by a ship’s captain in the 1840s. The journey she took to come into my posession is documented in letters that reside with her in her box, but I don’t know if the captain bought her new.
The oldest item I have that I got when it was new is a Zippy the Monkey stuffed animal whose age is 50. He long ago lost his yellow shirt, red corduroy pants and sailor hat, so he sits naked on a shelf in my bedroom. He is missing half an ear from the time I experimented with scissors, and has a blurry ink stain on his face from an occasion my brother was pissed at me.
I have a copy of Gunn’s Domestic Medicine from before Abraham Lincoln was president.
Ok, not the oldest, but old and very cool: I have a silk top hat from just after the turn of the century. It is the kind that compresses and then pops out. It belonged to my German grandfather and is in perfect condition.
Hey, the OP didn’t say! 
I’m not into old crap for the sake of it being old crap, except rocks. And especially Lake Superior agates.
If that’s the case, I’m made from some pretty old star stuff.
I have some decorative bird plates that belonged to my grandma, who died in 1963, but I don’t know how long she had them.
The oldest item that I’m sure of is a chair that an aunt gave my mom when my brother was born in 1942. It’s a miniature of a dining room chair, mahogany or walnut, with a carved rose. I’m not sure if it’s a true miniature, but it’s a kid-sized chair.
The oldest item that I don’t know the origin of is an iron (or some kind of metal) bell with the date 1808 engraved on it. My stepdad found it on a construction site in Seattle.
The oldest item I have that’s still in regular use is the wooden dresser I bought for $8 at Fran’s Antiques and Collectibles in Newton, Iowa back in 1977 (it was close to being an antique even then).
I refinished it, stripping off the hideous white paint and staining it a nice dark color. Still an attractive piece of furniture.
Now if I could only find a way to eradicate the trilobites from my basement…damn things are always scuttling around underfoot.
A deerskin page from an Egyptian Torah scroll from the 1600s.
I have a souvenir plastic coffee cup from the 1981 Major League Baseball All-Star game. I use it as a pencil & pen holder, which I’m certain has ruined its Antiques Roadshow retail value. 
That is an opera hat – collapsible so a gentleman could stow it under his seat at the opera. It is very cool that you have one at all, much less one that belonged to a family member!
I have a glass lidded compote that belonged to a great great grandmother who was born in the early 1800s, so I’m saying the compote dates to maybe 1840 or so. It is not a very interesting or attractive piece, but I hang on to it because I am an indirect namesake of that great great grandmother. I was named after a great aunt who was named after her.