I specify “human made” so some joker doesn’t mention his or her pet rock. 
For myself, it would be a box of discarded pot sherds I brought back from working on an achaeological dig in Israel - some of them date to the late Bronze Age.
Wife’s diamond ring?
Me.
Some coins that date back to 70BC or so.
Arrowheads.
Roman coins. One is Hadrian; the others, I don’t know exactly how old they are.
A couple of leafs from the Geneva Bible, published in the early 1600’s.
Eastern Woodland culture arrowhead, possibly 2000 BC.
AFAIK, a Winchester Model 92 rifle (not carbine) with the 24" octagonal barrel, in .32-20 Winchester caliber, that was made in 1897.
Now that I think of it I don’t have much that is old. I’ll probably inherit a few things from my mother but hopefully not for a long time. Maybe the oldest thing I have is my chunk of the Berlin Wall.
I own some arrowheads I found in the Ozarks but wouldn’t know how to date them. My kid has some Roman stuff (small buckle, coin) but the Romans were around for a long time so, again, I don’t know how old they are or if they predate my arrowheads. Either way, that’s the oldest stuff.
Me too.
A pastry blender my grandmother gave me that was originally purchased by her great-grandmother. I’m not sure how old that makes it specifically, but while it’s neat that it’s old, I mostly like it for the fact that it’s not metal wires in a plastic handle like my cheap one and does a far better job of blending pastry.
Arrowheads, stone tools, stone figurines, Greek coins and pottery shards.
And now for the opposite extreme:
I still own a few things from my college era, the mid/late 1970s. I doubt I’ve got anything made by man much before then. Damn near everything else I own is late 80s or newer, with the majority being post 2000.
I’ve certainly randomly had, and might even have now, some loose change from the 1940s-1960s. But I haven’t specifically kept any due to its age.
Arrowheads, a tomahawk my Dad found plowing the family farm.
Per LSLGuy, I still have a pair of Converse running shoes I remember purchasing about 1980 and are still wearable. They were originally metallic gold nylon and burnt orange leather! 
I have two wooden chairs that crossed the Atlantic with my maternal ancestors who came to America very early on. They didn’t cross on the Mayflower, but they weren’t far behind.
I have to admit, that’s pretty awesome. 
The oldest that I can reliably date are a few books over a century old. But I’m pretty sure I have some Indian stone arrowheads around somewhere, and might have some bits of old ironmongery too.
I have several thousand-year-old Chinese coins.
That’s not as impressive as it sounds; Chinese coils are made of bronze and were used a legal tender for centuries. I picked mine up for $1 each, which means I overpaid.