There's a couple ways.

If you are on Android, you can run Citrine and tell it to "Download your events" which will result in the app reaching out to a bunch of other relays on the network looking for any posts that you have made and download them locally to Citrine's database. Then you can export them by selecting "Export database" and do whatever you want with them.

You can also use a web tool: https://nostrsync.vercel.app/

This will find all your notes, download them locally to your PC, and then broadcast them to major relays on the network.

That said, when we talk about your data belonging to you, we mean that anything you post has been cryptographically signed by your private key, and it cannot be altered without it being obvious to every Nostr client that the post's content had been changed. Likewise, no one can post a fake note saying that it was you who posted it, because all posts require a signature, and their signature won't match, since they don't have your private key.

Who "owns" your data in the sense of where it is stored depends on whether you run your own relay, or you just use someone else's relay.

It's a similar concept to Bitcoin, actually. So long as you hold the private key that can be used to transact your Bitcoin, then you are the owner, regardless of whether you also run a Bitcoin node. However, if you don't run a node, you have to use someone else's node to transact. Same thing applies to use of private keys and relays on Nostr. The difference is, unlike Bitcoin where all nodes must store all transactions done on-chain, Nostr relays have no such requirement. A relay operator could choose to restrict the use of their relay to just their own notes, just notes from a circle of friends, just notes from users who have paid them, or any other rule they want to set about what sorts of notes their relay will accept.