I've been running a NC server for a couple of years now.
This is more complicated than it sounds - if you leave the server in sleep mode it may be possible to wake it with a "magic packet" ping, which comes with a new layer of complexity if you want it to be automated. I gave up in the end and just leave mine running - the little power draw to keep the little PC on seemed worth the trouble saved.
You can access it from anywhere if you have the server open to the internet. ( ie port forwarding through your router) . This gives you just a raw IP address on the open-side ( http://123.456.12.34 for example) which has all sorts of problems from technical to security: many services won't work with just an IP address, and without https protocol your connection is vulnerable to malicious activity. I started with just a raw IP address, but after some time shelled out for a domain name (10EUR/yr). This allowed me to access all services like music streaming, and to install security certs for proper https.
It sounds like a lot, but you can incrementally upgrade your system step by step and learn as you go.