Short answer: It's not technically possible.
Long answer: It's completely possible & occurs frequently (both through express & implied contracts).

Technically, no human can own another human. In good BDSM contracts (express contracts), you have a contract for the services of the submissive--which can be as expansive in function as ownership might seem. But it's still a contract nonetheless--it CAN be broken. How? Simple--either party breaches it. It can lead to different remedies for the non-breaching party, depending on what's written in, etc, but it's still a contract.

If you don't go the contract route & you go the "He owns me because He does" route, then you have an implied contract between you both for your services as a submissive/slave (agreed upon the material terms of the contract, etc). It still isn't "ownership" in the legal sense, but more of an exchange of services.

Now, substantively, many subs will say, "I am owned." This stems from their desire to be in an ownership--property type of relationship with their Dominant. It is more about the dynamics of the relationship then about its legal character.

Make sense?