Perhaps not own in the "i can sell it" way. I think some number of high schoolers that have never had their lockers searched, including myself many years ago, assumed their lockers ensured a level of privacy.

So... if I leave my house, am I subject to a warrantless search? (The road is government property, after all.)

The answer is no, I'm not (at least in theory; there are so many exceptions that you really are).

Moreover, you're not required to enter "other people's (or govt) property". Kids are required to go to school.

The reality is that students have been shafted by the Supreme Court ignoring what the Constitution says.