RAID 5 failure: 2 of 4 disks failed.
 

Is it possible to recover from this, whether automatically or manually? I know I won't get everything that was on there, but even a partial recovery would be okay. I'm not sure if they're really bad, or if they had small errors and may be able to work again.

RAID 5 means 1 redundant part (drive, partition or whatever) out of X. Loss of 2+ drives means the array is lost. All that's left is partial stripes (which means a data recovery service may be able to recover some small files).

Your only chance is to restore one of the drives to at least a semi-working state and force the array online. You may then be able to rescue some of the data. Whenever you hit a bad sector, the entire array would go offline again and the procedure to get it back online would have to be repeated.