I just encountered this problem. I clicked the Login button from the Shopify Homepage, and was able to login and get to my account settings, but when I tried to go to my Store admin page I got the dreaded "Due to an unexpected technical problem, Shopify is temporarily unavailable. Please check back in a few minutes - we'll be up and running in no time!".
I tried clearing cache and cookies, but still the issue persisted. I then tried on a computer that I had NEVER logged into Shopify with before and still had the issue. This tells us 100% the issue is NOT on the users side but on the Shopify side.
Now here is where the new insight comes in. The problem started because I was logged into my store and editing pages when a storm knocked my power out. Once power was restored, and I tried to log back in, this issue surfaced. It is my theory that when I lost power I was disconnected mid-session so when I tried to log in again, the Shopify servers already showed me in an active session and blocked the login. I had to wait some time for the session to expire before logging back in.
If I am right, Shopify needs to fix this. They should expire inactive sessions if someone tries to login with valid credentials. Say if the session has been inactive for 5 minutes (maybe 10 or 15) and a new login is detected the force end the inactive session and allow the login.
How you (maybe) can prevent this happening to you (unless there is a power outage). My issue was caused by a power outage, but I know many other people have encountered this issue without a power outage happening. I suspect that in those cases what occurred was that in their previous session they closed their browser or tab without manually logging out first. So their computer thought they were logged out, but the session was still active on the Shopify side the next time they tried to login. So always manually log out before closing your browser or the tab where you are logged into Shopify.
I could be completely wrong, but based on my observations and tests, this is where it seems to point.
Please respond here if you get this error and you 100% know you manually logged out of your previous session. Maybe we can get this figured out.