I would like to download the ReactOS Kernel Source Code for a project (because Windows_NT is closed source). I just want the kernel source, and nothing else.
The ReactOS kernel is named ntoskrnl. You can check it out (using SVN) on https://svn.reactos.org/reactos/trunk/reactos/ntoskrnl/ Note that the kernel is strictly tied to the Hardware Abstraction Layer (HAL) which you might need as well: https://svn.reactos.org/reactos/trunk/reactos/hal/
This will allow you to download ReactOS and explore offline its source code. However, you won't be able to build it: you will lack dependencies, tools required for compilation, and various configuration files. The proper way to build ReactOS is described here: https://www.reactos.org/wiki/Building_ReactOS. You need a full source checkout to be able to proceed, along with the official ReactOS Build Environment (RosBE).
If you just want to browse it (no build), then I'd recommend you do it online, either by using: http://code.reactos.org/, https://git.reactos.org/?p=reactos.git;a=summary, https://svn.reactos.org/svn/reactos/trunk/reactos/ or http://doxygen.reactos.org/. So that you're sure to always look at the latest version of the code available.