I was working on configuring a new database for backup to ZDLRA and hit this issue while testing a controlfile backup via Enterprise Manager -> Schedule backup. It could happen in any environment.

Unable to connect to the database with SQLPlus, either because the database is down or due to an environment issue such as incorrectly specified...
If the database is up, check the database target monitoring properties and verify that the Oracle Home value is correct.

The 2nd line clearly tells the problem but since the Cluster Database status in EM was green, so it took me a while to figure it out. Issue turned out to be a missing / in the end of ORACLE_HOME specified in monitoring configuration of the cluster database. The DB home specified was /u01/app/oracle/product/11.2.0.4/dbhome_1 instead of /u01/app/oracle/product/11.2.0.4/dbhome_1/.

On the server the bash_profile has the home set as /u01/app/oracle/product/11.2.0.4/dbhome_1. When I tried to connect as sysdba there, it gave an error TNS : lost contact. Then I set the environment with .oraenv and I was able to connect. /etc/oratab had correct home specified as /u01/app/oracle/product/11.2.0.4/dbhome_1/. After comparing the value of ORACLE_HOME in these two cases, the issue was identified. Then I updated the ORACLE_HOME value in the target monitoring configuration in Enterprise Manager and it worked as expected.