First, make sure wine and winetricks are installed. If you're not sure how to do this, then either follow one of the myriad guides available on the intertubes, or just run the commands below, which should work perfectly on Ubuntu:

sudo dpkg --add-architecture i386 
wget -nc https://dl.winehq.org/wine-builds/Release.key
sudo apt-add-repository https://dl.winehq.org/wine-builds/ubuntu/
sudo apt update
sudo apt install wine-stable winehq-stable
sudo apt install winetricks

After you've made sure that wine and winetricks are properly installed, you'll want to cd into the directory where you want BYOND to be installed; or open that directory in the terminal via some other means.

Finally, enter the following command:

curl https://raw.githubusercontent.com/Zero-Bits/ss13.sh/master/ss13.sh | bash

Run the runss13.sh script that was generated in your BYOND directory.

You can do this by cding into the directory and typing bash runss13.sh.

If this script isn't your cup of tea or doesn't work for you, then there are a few other methods of running BYOND on Linux:

1. Set everything up manually because you're a Linux wiz. (and/or know how to use wine/winetricks)
2. Use Lutris (https://lutris.net/games/byond/)
3. Play on a VM.
4. Play on Windows.

Credit to /u/Xaltonon on reddit for the original bash script and guide, as well as /u/KarniinTheMighty on reddit and Gman0064 on github for their fixes and guides.

Additional credit to Zamura-0001 on BYOND for the icon being used.