What security guarantees do blockchains provide for data storage? When it's considered, like here

a project where a user needs to store a (small piece of data) on a blockchain

what does that really mean from a cryptographer's perspective? What security guarantees does it give? Things like availability (to which entity, under what condition and delay, with selection according to what criteria/tag), proof of origin (for one who finds that data), non-repudiation.
Is there a formal definition of a blockchain's expected properties, like we have for e.g. PRG, or a key exchange protocol?
Update: Emmanuelle Anceaume, Antonella del Pozzo,  Romaric Ludinard, Maria Potop-Butucaru, Sara Tucci-Piergiovanni's Blockchain Abstract Data Type attempt that. But it flies too high above my head. In retrospect I should have first asked: what's a blockchain, with a simple example?

Update: these NIST slides give a visual illustration of a minimalist blockchain:

They also make a basic but essential distinction between

Permissionless Blockchains, decentralized and often tied to cryptocurrencies.
Permissioned Blockchains, involving trusted participants or perhaps an authority.
What security guarantees does it give? Things like availability (to which entity, under what condition and delay, with selection according to what criteria/tag), proof of origin (for one who finds that data), non-repudiation.

In respect of data storage the blockchain itself (in a decentralised form) gives us very few guarantees.  When it is coupled with a cryptocurrency (eg. Ehtereum) then it isn't the blockchain that provides security primarily, but the cryptocurrency via the blockchain. In such cases it is the miners of the cryptocurrency that have a vested interest to keep your data.  And it costs money.
The blockchain is useful as a means to deliver something worthwhile.  Much like torrents if there is no one seeding, then it isn't a useful means of delivery.