There is no mining of Storj.   You can rent hard drive space and earn some tokens from the Storj Labs team. 

SJCX was the old Bitcoin based token, but due to high fees and a desire to work with smart contracts, Storj Labs transitioned to the STORJ token.   

In terms of investing for equity in the company, that is not currently an option, though at some point the plan is likely to sell shares.  Since they had a Pre-Series A run at BNKtoTheFuture some time ago.  But right now, nothing of the sort is going on.

Storj Labs is currently working on major improvements to the protocol as well as the payout scheme.  Details are updated in their blog posts.