I'd like to have my own server at home sorta like a home AWS.
How to set up one and make it available to anyone over the Internet? What tech specs should I buy (RAM, CPU, # of cores, operating system, etc.)?
How much does it cost to keep one running all the time?
Be extremely careful. Plenty of people are really smart and malicious, so you need to isolate it from everything on your network. You're giving random people remote code execution on your local network, which is like the worst case scenario for security.