Q: iPhone 4 will not restore (Error 4005).

Are you using a USB port directly on the computer, not a hub? If it's a tower or desktop, are you using a back port, not a front port?

Have you tried temporarily disabling your antivirus and firewall?

That's likely, but try a different computer, and take it to an Apple store and let them try. Make an appointment first, as they are pretty busy.

I too am having this problem with my iPhone 4. First code popped up was 21, then it was 3194, now it's saying 4005.

You can see the tips for all of those errors here: http://support.apple.com/kb/TS3694

3194 usually means you are trying to install a version of iOS that is different from the current version (7.0.3). It, 21 and 4005 can also mean that your antivirus is interfering with the update. As that is the common denominator, it is the one I would check out first.

I've been in contact with Apple Support already and they haven't helped much. I keep getting 21 and can't restore at all. It just errors then loops apple logo. I swear, I don't have any antivirus software installed on the entire computer.

Running Windows 8.

I had the exact same.
What I did to fix it:

Signed out of iCloud on PC.
Tried again -> Done!