You already cannot mine for yourself right now, every mine is owned.

Every mine is owned because a mine is not a naturally occurring object. They don't exist until they're built, at which point someone owns them. You can always go start your own mine if you want to invest/risk the time and resources. This is, in fact, a great example of why capital matters: if you invest your talent in discerning a location for a mine in a place where nobody else has a claim, your blood sweat and tears in constructing the thing, and capital in the form of tools and feeding yourself while you work prior to making a return, all at the risk that the mine you're building never produces a thing, you are entitled to reap the ongoing benefits of the thing you built, at great risk to yourself, even if you hire others to help you with your efforts after the mine starts paying out.

Actually, it's not even quite true that "every mine is owned". I've personally been to several abandoned mines in multiple Western States. Under ancap ideology, you could begin mining one of those. Just because some guys with 19th century tech couldn't make a profitable mine doesn't mean there's nothing there.