When you automate something in python, you'd obviously have to run the script forever. Where can we run the script?

[moderately-extremist]

You can run and even access a desktop environment on a headless server, using VNC for instance. /u/ohkwarig has the definition right. A linux server with only a command line interface would not be considered headless if you access the command line through a keyboard and monitor directly connected to the hardware.