To simply answer the question how can someone own you… you let them. But it is in fact the answer to the question is a much more complicated then that. The answer to your question how can someone own you varies. And that variation all depends on the personal definition of those who are owned and who do own someone. To me when someone says that their Master/Dom owns them, what they mean by it is that they have submitted themselves in such a way that they now belong to that person. They have surrender control over their mind, sprit and body to that person. And in this sense they have given ownership over themselves to their Dom/Master.
When I read your questions it also seemed like you were asking how someone can call themselves owned. How can one allow them to be owned? Now, I’m still very new to the BDSM lifestyle myself… But to answer these last two questions is trust. Almost everyone you see who say that they are owned by someone, don’t just let anyone claim ownership over them. The process takes a lot of time and more trust then almost all other relationships take. They trust their Dom/Master not to ever abuse the power they have given him/her by surrendering everything to them. And to have someone trust you so implicitly is not something they would ever want to abuse. Now, in most of the relationships you’ll see the Dom who owns a sub does not make every decision in his/hers life. Partly, because that is impractical and partly because they don’t have the desire to control their subs life in every minute aspect. But in the end they have the power to control those aspects because their sub has surrendered themselves so completely to their Dom/Master that they now own them.