Alternate Xcode Command Line Tools Installation Methods

If you cannot install the Xcode Command Line Tools from within the app, there are a few alternate methods that you can use to install them.

Method 1: The command line

Open Terminal.app (located at /Applications/Utilities/Terminal.app)
Type xcode-select --install and press return
In the window that appears, press "Install"
Wait for the command line tools to download and install. It should take somewhere around 5-10 minutes, depending in your internet connection.

NOTE: When you start the download for Xcode Command Line Tools it may first estimate the time remaining at several hours or days. This is normal, it will typically not take anywhere near that time (speed will vary based on your Internet connection). If you're curious why this happens, watch this video by Tom Scott.
If you get an error, proceed to method 2

Open the Apple Developer website in your favorite web browser
Click the "Account" button, located in the top right
Sign in with your Apple ID
If asked, go through the Apple Developer Account setup until you reach a page that looks like the one pictured below. DO NOT START TO JOIN THE APPLE DEVELOPER PROGRAM, doing so will place a lock on your developer account, and you will either have to pay $100 or wait an indeterminate amount of time for your account to be unlocked
Go to this link
Select the correct download, click on more details, and download the .dmg file a. If you are running a release version of macOS (non-beta), select the most recent Command line tool file that does not have the words "beta" or "Release Candidate" in its name b. If you are running a beta version of macOS, select the most recent file with the word "beta" in it' name.