One way to view Storj is as a follow-on to IPFS. IPFS's design assumes people aren't willing to store data for each other, and aren't willing to entrust their data to others' servers. Storj aims to reverse these assumptions with mechanisms to pay for storage, to verify that servers are storing what they ought to, and to maintain replication as storage servers fail.
Would you entrust the only copies of important data to Storj? Would you be willing to store and serve other people's data, e.g their pirated movies?
In what ways is Storj better than existing cloud storage, such as Amazon S3? Is it likely to be cheaper? More robust? More secure? Higher performance?
If Alice stores a file in Storj, and wants Bob to be able to read the file, what information should Alice e-mail to Bob (a file name? a key?)? What does Bob's client do with that information in order to retrieve the file? That is, what steps does Bob's client (and Satellite) take along the way to retrieving the file's content?
Storj encrypts stored data. How does Bob's client obtain the key needed to decrypt Alice's data?
Each Storj user has a Satellite service which they must trust. Does that make sense, given that part of the point is to exploit untrusted storage servers? Why is the Satellite necessary?
Is it a problem that Alice and Bob may use different Satellites? Can they nevertheless share files?
Could a storage server cheat on an audit, i.e. pass the audit without actually storing the data?
If you have time and interest, it's worth comparing Storj with Filecoin, a competing project by the people responsible for IPFS.
The questions below are only relevant for the 2016 version of the Storj white-paper.
How does proof-of-retrievability work?
What is the purpose of the pre-leaves?
Is it the case that only the owning client can verify storage of that client's data? Is that a problem?
What state must a client store in order to verify that others are indeed storing the client's data?
Can anyone other than the owner retrieve data? If yes, are there ACLs or other access control? Is there shared read/write data?
What good are the contracts? Are they enforceable? Is there a penalty for breaking a contract?
How do Storj's market, payment, and verification mechanisms compare with Filecoin's?