If someone owns a piece of land, how deep does their ownership go? Do they own it all the way to the center of the earth?

It depends on the property.  There are often easements and other things that give you rights to footage on the neighbor's property.  You might also have rights on a shared driveway.  Regarding the actual land depth, you own it as deep as deep is, less any mineral rights and other things as outlined in the deed. 

Not very deep! And definitely not to the centre of the earth, and all the way out into space. Generally, you don't buy the mineral and oil rights or even the treasure rights. Those are reserved by the State (country), but you may have the right to bore down to an aquifer for your own private use.

I think it goes pretty deep but at a certain depth, unless you get oil or archeological treasure, it's just all liability and no good thing comes from it. 

That really depends on the mineral rights if you own the mineral rights you have the right to dig down as deep as you want to dig down with a permit to dig a well permit excavation permit or mining permit. You own the land all the way to the center of the Earth but accessing that land will require permits if you intend to actually start digging.

Technically, yes.   But that doesn't mean you can do anything you want with it.   Just like with above-ground, there are land-use regulations for things like clearing, excavating, etc. and then there's issues such as water rights, mineral rights, natural gas rights...

So yes, even though you *own* land, your rights to use it as you see fit have legal restrictions - both above and below ground.