From the error log, there are failed logins, are you trying to connect using any of the below ones ?
Login failed for user 'drily'. Reason: Could not find a login matching the name provided. [CLIENT: 172.17.0.210]
Login failed for user 'WCC\baadmin'. Reason: Failed to open the explicitly specified database. [CLIENT: <local machine>]
Login failed for user 'sa'. Reason: Password did not match that for the login provided. [CLIENT: <local machine>]
Also, from your screenshot, the report server databases are cleanly shutdown. Do you have auto close option turned ON ?
If you have rights on the server, then below TSQL will help you :
'RESTORE DATABASE <dbname> WITH RECOVERY' or 'ALTER DATABASE <dbname> ONLINE'
What is the output for below SQL:
SELECT name, user_access_desc, is_read_only, state_desc, recovery_model_desc, is_auto_close_on, is_auto_shrink_on
FROM sys.databases;
UPDATE:
I see the problem now ... that auto_close option is turned ON for report server databases. According to BOL:
When set to ON, the database is shut down cleanly and its resources are freed after the last user exits. The database automatically reopens when a user tries to use the database again.
When set to OFF, the database remains open after the last user exits.
Refer to herehere and herehere for more detailed explanation.