naturally, the local system account does not have access to network shares unless you have granted full access to any account, which i shouldnt suggest. I believe in a domain environment you have to give the machine account of the host privileges to that share to. I would not recommend this. For a domain, use an domain administrator account or create a backupadmin account for this sole purpose. For a workgroup environment, make sure to use an account that has full read/write access to the network share or naturally you will have trouble.
The errors you get in your log would logically suggest an issue with your hardware. That is, either u get timeouts with the controller driver / hardware when reading data or there are physical defects on your disk. However, that could not be the case if you do not experience this behaviour by simply changing to a user that is a member of the local administrators group.
What OS are you running? What types of drives and on what controller?
Normally and in almost all cases, the I/O Device error message in eventviewer is something that is reported from your hardware, in most cases your harddrive or your harddrives controller driver.
A very basic, but simple advise worth trying. 1; if you have Antivirus products running, temporarily disable it. 2: make sure you have the latest drivers installed for your HD controller.