Apparently, there are issues with your I/O subsystem. You need to resolve these before you start thinking about Transactional replication. Since DBCC comes out clean, there is some hope. But it is bad that backups are failing, since you most likely need to move the database to new hardware. Then again, that depends on what those disk actually are. Locally attached disk? Disks on a SAN? Virtual disks on a VM host.
The best is if you have a clean backup of this database before this started to fail, and then can apply log backups. Do this on a different server. Take the applications that are running against these databases offline, as keeping them running could be a gamble. You could lose data, if the database becomes corrupt.