Windows 10 doesn't download updates and gives me error 0x800703f1

It sounds like your computer needs a reinstall... though I'm looking for a way to do it without loosing your files.

If you want to start from scratch with a clean install even with the free 10 upgrade, see this: Windows 10 - Clean Install - Windows 10 Forums or this: Refresh Windows 10 - Windows 10 Forums - these will remove all files but it will likely fix the issue.

Keep reading for another potentially easier option below-

HERE WE GO: try this to keep your stuff but possibly fix the issue, so you don't have to completely reinstall: Refresh Windows 10 - Windows 10 Forums

The last few things I see that may be beneficial to you. I have nothing more. Please keep me posted if you decide to go with one of the options and how it turns out.