This is effectively the idea behind the Ethereum blockchain model. You can make "smart contracts" (think APIs with persistent state) using languages like Solidity[1] and deploy them onto the blockchain. After that, you can invoke individual functions by paying "gas" (small amounts of ETH) that goes to the node operator's account. Smart contracts also get their own blockchain addresses, so they're capable of sending and receiving transactions. Meaning you can build financial applications with little to no barriers to entry! (whether this is a good thing or not, I will leave up to the reader).
The really cool thing about it is that nobody owns a smart contract once it's deployed. You can't edit the code or even delete the contract itself. It's a truly autonomous entity that will continue to operate the same way forever (unless there's a 51% attack or something of that nature).
The obvious benefit for this is that you can mathematically ensure trust. For example, if you hosted a lottery app on a LAMP stack, you could steal the money, hackers could get into your server, your database could get corrupted, etc. On the blockchain, nobody can access your lottery funds or business logic, not even yourself, meaning that as long as you developed the application correctly (to be fair, that is a big assumption), it is truly fair.
[1] https://solidity.readthedocs.io/en/v0.5.10/