Google Compute engine has a free tier, you still give them your credit card, but when running one of the lowest combination of linux VM/CPU/ram 24/7 the discounts actually add up to $0.00 a month.
A raspberry Pi costs about $40. Can run 24/7 at home. You can remote into in using VNC from your local network, or SSH into it. If you forwarded an SSH port through your router you can use an SSH tunnel. If you wanted to securely access it from anywhere using a terminal program like putty, with VNC and an SSH tunnel you can even remotely and securely control the GUI.
Edit: Oh I forgot to mention your IP address, use something like duckdns.org on your pi to make sure you can still get to your IP address from anywhere in the world.