in the official requeriments (Supported Graphics Drivers on Linux | Unreal Engine Documentation) they don’t mention HD space, but what i can tell you is: epic games launcher + unreal 4.13.2 uses 20.9 GB in my computer, i have a custom version of the engine in my computer that uses 48GB, my projects uses from 5GB to 10 GB, so you will need at least something around 30GB to install and create a project, you may want to download some contents from the asset store or from the “learning” tab, so have another extra 10GB - 20GB, also it’s recomended to have some extra disk space, if you use 100% of your disk you surely going to have some problems.
so i would say from 30GB to 50GB, also this isn’t a exact value, it may be different in your pc due to disk fragmentation.
also, i’m saying about windows (in my case i’m using windows 7), this may also be different in mac and linux.
ps: before installing unreal engine it calculates and tell you how much space it needs in disk, just the engine (with just windows as target platform, without source code and without starter content etc.) it will use 10,46 GB of HD, while download the engine with everything will use 29GB.