Erased Hard Drive, can't reinstall OS X El Capitan.
Preface: I was able to answer my own question, but it took me a very long time to do.
As such, I'm posting this not as a question, but as a guide for anyone who needs the help.
The Problem: I recently erased the two hard drives in my Mac Pro from about 10 years ago. When I started the computer, after choosing my language, I was sent to "OS X Utilities". I presumed I need to "Reinstall OS X", but when I selected it, I clicked "Continue" to set up installation of OS X 10.11, where a dialogue window appeared and read: "To download and restore OS X, your computer's eligibility will be verified with Apple." I clicked "Continue", but an error dialogue appeared reading: "A required download is missing." After browsing forums for hours, I noticed most of them say that you need to create a bootable installer, but none of them acknowledge the fact that both hard drives are empty, and that I couldn't access finder, let alone download anything from the internet. Apple support pages assumed I had the installer in my Applications folder, but I couldn't access my recently-erased Applications folder...
The Solution: What other posts did not make clear is that you must download the correct installer (Found Here) on a completely separate Mac. After downloading the right installer, find an external drive that you are comfortable completely erasing, and plug it into the Mac with the installer. You then use the command found Here in Mac's program Terminal, to transfer the installer to your external drive. (Be sure you replace MyVolume with the name of your drive.) When terminal has completed - which may take half an hour or more - your external drive will be renamed, and can be ejected. Insert the new boot drive into the Mac you want to install the OS on. Press and hold the option (alt) key immediately upon starting your computer, until a screen appears asking which drive you'd like to boot from. Select your new boot drive and follow the prompts on the computer. Eventually, you will be brought to the OS X Utilities screen; select "Reinstall OS X" from the menu and follow the prompts to install your OS. It took the better part of an hour for my computer to get to the point where it automatically restarted, so some waiting may be required. After the automatic restart, the install will begin. This took around half an hour for my computer. At this point I was able to follow the prompts to finish the setup and that was it!
Epilogue: My biggest failure was not understanding the need and function of a second computer. Trying to make Terminal do things on the 'broken' computer wasn't effective for obvious reasons. I didn't realize that I needed to run the program I downloaded on the first computer (though, this may be exclusive to El Capitan) before running the Terminal command. All in all, this may have been an issue of my own incompetence, but I got it done. I hope this is able to help someone else.