Similar to the issue I had previously installing .NET 3.5 on Server 2012, the same problem manifests in Windows 8/10 and later where the PC tries, and fails, to use WSUS to get the requisite update files.
Making a small change to the Local Group Policy will help get it installed:
1. Run gpedit.msc
2. Browse to Computer Configuration | Administrative Templates | System
3. On the right-pane, double-click on ‘Specify settings for optional component installation and component repair’
4. Set this to ‘Enabled’ and also enable the check box headed ‘Contact Windows Update directly to download…’
No need for a reboot, .NET 3.5 should then be downloadable.
[EDIT – 23/05/2016] If, as I’ve just encountered, despite the change above it still won’t download, and if you have the OS source (DVD or ISO) available, then you can insert/mount the source disk and then run the following command from an elevated command prompt:
Dism /online /enable-feature /featurename:NetFX3 /All /Source:D:\sources\sxs /LimitAccess
Replace the drive letter with that of your drive/mount point of course.