I'm wondering, in looking at your two statements, the error message:
A nonrecoverable I/O error occurred on file "E:\SQLLogs\MyDB\MyDB.ldf:" 1(Incorrect function.).
Then your backup code:
BACKUP LOG [MyDB] TO DISK = 'M:\Backups\DataCap\MyDB_log_20130102.bak' WITH RETAINDAYS = 7, INIT
If possibly your log may be on a bad sector of the HD? Maybe, if possible, try moving the log file (temporarily) to another location, ideally another drive entirely (as I would suspect the OS would just change the pointer in the file tables on the disk to point to the new location, leaving it on the bad sector.)
Of course, the trick is both getting the time to do this, and then doing it. Although also, chkdsk *should* have found bad sectors when you ran it (at least reported them, I would NOT tell it to fix them without having backups of everything on the disk, it could end up being like running dbcc checkdb REPAIR_ALLOW_DATA_LOSS...)
We would occasionally run into the "Incorrect Function" problem when we had backups (not SQL) going to CD-RWs and the drive or media was going bad.
Also, as opc.three asked, what about in the Windows Event Logs around the time you tried taking the backup? Especially the System log.
Jason