You have a private key and a public key and optionally you could password protect your 'account' after which all transactions would require you to enter this password. This password would obviously be stored on the blockchain itself.
Is something like this possible to implement?
No. For this to be possible, someone has to validate that your password is correct. They can't do it without the password hash or password. While it could be possible to store the passwords in the nodes, the number of passwords would be too big. After all, the user is supposed to be responsible for his own security and its not someone else's.
When you spend any outputs from your address, you would have to expose your plain password to everyone. Anyways, if you can get your private key stolen, your password would probably be stolen with it.
Service like TrustedCoin uses 2FA and multisig to implement similar ideas. They do cost money though.
While it's not same with what you mentioned, it's possible if you use OP_RETURN function.
You need use BIP38 to encrypt your private and add your encrypted private key to a transaction with OP_RETURN, even though OP_RETURN function isn't supposed to store any data besides transaction/P2SH for features such as multi-sig. Also, it's risky to expose your encrypted private key, even if you encrypt it with strong passphrase.
You also need to remember the transaction ID and the process isn't user friendly.
What for though? I don't think the OP wants a method to store his encrypted private key publicly. It should never be shared online, much less a platform for which everyone can see it.