When opening Terminal: "NO LOGINS: System going down at 01:10"

I don't know why, but I resolved it by deleting the file nologin inside the /etc/ folder.

Perhaps here is some explanation, from the book A Practical Guide to UNIX for Mac OS X Users, by Mark G. Sobell and Peter Seebach:

The pam_nologin module makes sure that if the /etc/nologin file exists, only the root user is allowed to log in. (That is, the pam_nologin module reports success only if /etc/nologin does not exist or if the root user is logging in.) Thus, when a shutdown has been scheduled in the near future, the system administrator can keep users from logging in on the system only to experience a shutdown moments later.