Why does .NET Framework 3.5 refuse to install on my Windows 10 PC?

Why does Windows 10 keep failing to install .NET Framework 3.5, whether I try it through Windows Features or download the offline installer manually? It always stops with error code 0x800F081F and says the source files couldn't be found. I've already tried running DISM with the /Online /Enable-Feature /FeatureName:NetFx3 /All /Source command pointing to the Windows ISO, but it still fails every time.

Could this be caused by a corrupted component store, a missing side-by-side folder, or a group policy blocking the installation? And is there any way to get .NET 3.5 installed without doing a full repair install of Windows?