Are you editing out the filenames in those messages, or is SQL really reporting extensions without names?

Operating system error 1450: http://blogs.msdn.com/b/psssql/archive/2008/07/10/sql-server-reports-operating-system-error-1450-or-1452-or-665-retries.aspx, so you may not be able to use a snapshot at all, if the data changes are exceeding what's allowed for a snapshot

Can you make enough downtime to run an 'offline' checkDB?

If so, take the DB into single user mode and run the following (exactly the following please)

DBCC CheckDB ('<database name>') WITH NO_INFOMSGS, ALL_ERRORMSGS, TABLOCK

Mission critical database, no backups, long term corruption problems? :crazy:

Repeated corruption indicates a problem with the IO subsystem. Doesn't necessarily mean the disks, could be anything in the IO stack from filter drivers right the way down.

If that was my DB, I'd be looking to get it onto alternate storage ASAP (even if that means performance degradation) and getting a backup taken by any means necessary. As it is, you stand a fair chance of losing that DB entirely. Seen it happen.

Fixing the corruption is not your primary goal here, getting the DB onto stable storage is. Otherwise you could repair and be back in this spot tomorrow.