Cloud computing

Is it possible to use the blockchain to run computation on a network of computers? Like Storj but the operating system / CPU is running on a bunch of free computers in a network?

The simple answer is no because storj has nothing to do with a blockchain, Also the storagenodes have no computing what so ever other then sending and receiving data for storj network.

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The blockchain is only immutable database, so it cannot be used for computing anything except what designed for that blockchain (usually - the next block and validation). It can store only small chunks of data. There are decentralized computation platforms though, which is able to provide smart contract functionality, but it has limited usage (usually only related to the same blockchain).
But there are many companies trying to build a decentralized computation platform, which could work as a cloud computing: Golem, Flux, SONM, Akash, Ethernity, Iex, …
But all of them did not solve the problem with security of customer’s data and processes. I, as an operator of such decentralized computation can interfere to the customer’s process and can get an access to their data. So, nothing similar to Storj in that place regarding security of your data.

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Right, you’d need some kind of sandboxed environment on machines that prevented the host from being able to access the sandbox. It may be possible but would not be efficient if you were going to mask the processing and transfer input/output encrypted. If you aren’t concerned about data privacy, there are already things like all the clustered computer for @home projects that are out there. I think the idea for a lot of those was that people could get paid by companies who needed a lot of processing power and could rent their CPU time. But, as far as I’m aware, the amount companies were willing to pay for it didn’t meet projections and almost all of it is donated processing time now versus paid.