Most Linux desktop environments are DPI independent for fonts and toolkit controls, but it can be a bit hard to change as such things are often tied to your system theme. Of course, that doesn't help with scaling things like images. For many years now you could get desktop scaling using Compiz, but that requires hardware with good OpenGL support so few distributions use it. The current standard for things like 4K monitors is HiDPI (which Apple is calling Retina for marketing reasons).
The only Linux distribution I know with good support for HiDPI is Linux Mint Cinnamon. It even selects it automatically if it detects that your monitor exceeds a certain number of pixels per inch. The setting is in Settings -> General -> Desktop Scaling. I find that with HiDPI and a some tweaks to the default fonts, only web browsers don't display how I want them to (I prefer a 110% zoom for my web browser). Fortunately, changing the default zoom in Chrome works very well, it can even scale Flash content properly.
Other desktop environments that use Gnome libraries like Unity and Gnome Shell should have HiDPI working soon (if they don't already). It looks like KDE has HiDPI support, but they still have some issues to resolve. I'd expect the new KDE 5 desktop to work well.