Yes, that's the correct machine. I cannot install Windows either, but that's because the Windows installation disk is badly scratched (I got the computer second hand). I have done md5 checks on all the ISOs I've been using, and burning them to disk with verification enabled.
I ran diagnostics on the CD/DVD Drive, Hard Drive, and memory, with no results. I couldn't reproduce the error while copying large files, however I could while recursively copying large directories, at least if the source of those directories were on the disk the OS was running from. In particular, I booted from a Sabayon live USB drive, then plugged in another USB drive, then opened the file browser as root and used it to copy /var into a subdirectory of the USB drive. When I did this, about 30 out of the thousands of files in /var failed to copy due to similar input/output errors. Interestingly, I had no similar problem when copying that same directory from the second USB drive to a third USB drive or to a partition on the hard drive. Perhaps the problem only occurs when copying files from a UnionFS source?
At any rate, I figured that the problem was not with any of my drives, but with the motherboard or the BIOS. Resetting CMOS did no good. Furthermore, eMachines does not provide BIOS updates or images for download. After much searching, I was able to find a BIOS update over at the Gateway website (http://support.gateway.com/s/MOTHERBD/Intel/4006158R/4006158Rnv.shtml). Unfortunately, Gateway only provided a self-extracting ZIP file which included a Windows program for flashing the BIOS (along with a BIOS image and some other data). I didn't have Windows installed on the machine at the time, and couldn't install it. Instead, I had to run the self-extracting archive on Wine, then pull the BIOS image out of it. Then, I downloaded an ISO image of a BIOS upgrade utility from Intel (the one here-> http://downloadcenter.intel.com/confirm.aspx?httpDown=http://downloadmirror.intel.com/14364/eng/CL0034P.ISO&lang=eng&Dwnldid=14364&OSFullname=%0a&ProductID=2613 is almost the right model number for my motherboard, but not quite), extracted the files, modified them to install the BIOS image I got from Gateway instead, created a new ISO using mkisofs, and then burned that to a CD and was able to use it to update the BIOS. Now it works fine. This also fixed the occasional problems with X.Org crashing that I was having.
TL;DR: The BIOS was faulty. With a lot of difficulty and hacking, I updated it. Problem solved.