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?

At common law you own your land from the center of the earth to the top of the sky. In reality, there may be superseding rights of others, by title or by statute.  In the USA, the FAA has exclusive rights to "airspace" at some level over your property, depending upon circumstances.  Your property title might also contain a transfer or other compromise of your rights to harvest underground minerals (or even gravel) from your property, let alone buried treasure rights.

There are other forms of underground and "air" rights. I have a 250kv powerline running above one of my properties due to their "easement". Other utilities may have the right to lay and maintain various pipelines or wiring underground. The good news is that the utility company is assessed property taxes on its equipment and pays about 95 percent of my property taxes.

I have property that lacks mineral rights, which were transferred to a mining company in 1842. It would cost me a few thousand dollars for the paperwork and legal fees in court to get those rights back, upon default of the successors to whatever company had the mineral rights.

Depends on if they control the mineral rights also.

Plus any easements on the property. 

All would be spelled  out in the purchase contract 

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.