Or the oldest thing in your home? Your oldest family heirloom? It fascinates me how we're physically connected to the past. Don't worry about whose individual property it is, that tends to be pretty vague.
I asked this on Musing once, but most users there were young people from the third world, who didn't own much of anything.
I'm not from a rich family either. In my case, the oldest thing I own is a book from 1885, or actually one yearly volume of a magazine called De Aarde en haar Volken (The Earth and its Peoples). It's older than my grandfather, who was born in 1899. In 1885, Karl Benz created his Motorwagen, but he didn't reveal it until the next year.
In the spirit of this group, as explained by @acidyo, I want to keep this an open question, so I wrote a separate post about my answer.
Sketches from Norway
The oldest text I have, counting modern editions but not translations, is the Roman history book Ab Urbe Condita by Livius, in Latin. I can't read this, although I recognize almost every word.