As of the moment, I am running a Windows 8.1 PC that does not have the storage or the RAM for a IDE like Android Studio or Eclipse. I want to download the Android SDK tools, without the IDE. How can this be accomplished?
You can find the command line tools at the downloads page under the "Command line tools only" section.
These are the links provided in the page as of now (version 26.1.1):
Windows no installer: https://dl.google.com/android/repository/sdk-tools-windows-4333796.zip
MacOSX: https://dl.google.com/android/repository/sdk-tools-darwin-4333796.zip
Linux: https://dl.google.com/android/repository/sdk-tools-linux-4333796.zip
Be sure to have read and agreed with the terms of service before downloading any of the command line tools.
The installer version for windows doesn't seem to be available any longer, this is the link for version 24.4.1: