By design, a public blockchain is an immutable record maintained by a decentralized network where no central authority has the power to go back and remove something from the record. This makes the blockchain an exciting technology for resisting censorship, but it also means that if someone adds your personal data to the blockchain, your options for removing it are very limited. You would need to shut down all nodes in the network or convince all nodes in the network to agree to a hard fork. In the latter case, nodes that don't join the fork would continue to have the private data.

That's true for unencrypted information stored directly on the blockchain. For applications where you need the ability to delete data and don't need strong censorship resistance, one solution is to store private data off-chain and only store the location and hash of the data in the blockchain. This article discusses that idea in detail:

https://medium.com/wearetheledger/the-blockchain-gdpr-parado...