For the database to go suspect or offline, an event occurred, probably something with hardware. When SQL Server brings the databases online, it performs a simple validation check. if they failed, they're marked as suspect. Basically you have potentially corrupt databases. The first thing you need to do is run DBCC CHECKDB('databasename') for each of your databases that is suspect. Based on the information that returns, you can try to figure out what to do. Do you have backups in place?
There are many, many reasons for a database to become marked "suspect", and it's not always because the database has become inconsistent. If you restart the SQL Server service, SQL Server will try to set the database online, and if it fails, it will mark it as suspect. If that happens, you can have a look in the ERRORLOG why SQL Server didn't manage to set it online. Databases having Auto Close set to 1 is one cause for randomly getting suspect databases. If a process accesses the database file while SQL Server is trying to set it online, it will be marked as suspect by SQL Server.
Ther may be many reasons for db in suspect/offline -->datafile or logfile may be inaccessible due to virus scan(generally we do) at the time of databases are staring up -->curruption in the data or log files. -->if the restor of large databases or in the middle and some one intentionally stop(kill) the process the process may be in hang state and the db status may show suspect. (some times we do resore using 3rd party tools. due to network issue cummunication may fail between server and the tool(tool may be in another server)) -->some one intentionally bring the DB offline Solution: As Magnus said check in the errorlog you may get some information related to why not the server bring the db online, run the dbcc checkdb you will get some what information. (you said that you are unable to run dbcc than what the error you are getting while running it),If you unable to run dbcc than try to backup the tail log (if log file is acessible) and resore with recent backups.