Install WSL Without Microsoft Store

Installing a WSL (Windows Subsystem for Linux) distro may not be possible on your current machine. This could be due to a corporate network policy or other restriction in your environment. Microsoft provides direct downloads to the Linux distros available for WSL so you can manually install them:
https://docs.microsoft.com/en-us/windows/wsl/install-manual

If you need to enable WSL in your Windows OS, open PowerShell as an Administrator and run the following command:

Enable-WindowsOptionalFeature -Online -FeatureName Microsoft-Windows-Subsystem-Linux

You will still have problems with installation if your system does not allow files with the ".appx" extension. To circumvent this issue, rename your downloaded file from the ".appx" extension to ".zip".
e.g. Ubuntu.1604.2014.711.0_v1.appx -> Ubuntu.1604.2014.711.0_v1.zip

Extract the contents of the zip file into a directory and run the ".exe" found inside.

Additional resources:
Microsoft - Manually download Windows Subsystem for Linux distro packages
GitHub - Distros fail to launch: File system error -2145103860
GitHub - WSLGit: This project helps integrate Jetbrains IDEs with WSL