Google has said time and time again that 404s are normal, they don't cause penalties or indexing issues. But a similar question came up about 410s, which are stricter versions of 404 status codes.
John Mueller said on Twitter that 410 status codes "generally don't cause issues with crawling." He added that you should just "mark as fixed is UI-only, it doesn't change indexing/crawling."
Of course, if the page that is returning the 410 or 404 is a page you want indexed, that is an issue. But outside of that, the 404s and 410s on your site, should not impact the crawl and indexing of the pages that return a 200 status code.
@VishMarathe411 No, these generally don't cause issues with crawling. Mark as fixed is UI-only, it doesn't change indexing/crawling.
— John Mueller (@JohnMu) August 24, 2016
Forum discussion at Twitter.
