Google's John Mueller said on Twitter that most of the 404 error codes that Google picks up on are not the webmasters fault. He said many people link weirdly or have broken code when they link, resulting in the 404.
Here is John's tweet:
@suzukik @jenstar Most 404s aren't the site's fault; people (+sites) link in the weirdest, broken ways. Hiding the errors doesn't seem right
— John ☆.o(≧▽≦)o.☆ (@JohnMu) January 3, 2017
So if you see 404s, don't take it personally, every site has them.
Forum discussion at Twitter.