While playing Story of Seasons: Pioneers of Olive Town, players are going to come across a lot of Timeworn Objects. These can be found in chests while digging for treasure, and while mining. A common source is the baddies that will be tunneling through the mines. If you can hit them on the head with your hammer, they may just drop a Timeworn Object for you.
You can also find objects in small ponds dotted all around the farm if you empty them out using your bucket.
They can contain all manner of rare and hard to find items, from clay and ores, to diamond rings worth 2500 Gold, which is a hefty sum for any struggling farmer. The only way to find out what is actually in these objects is to bring them to the museum.
Make your way to the town square and go into the large grey building on the right, then turn right inside the door. Head to the younger lady at the counter and speak with her and she will give you the option to appraise the items. She will check them out, randomly determining what they are.
After that, they are yours to do with as you will. Some can be sold, others kept as valuable resources, and some will even make perfect gifts for the right person at the right time.