Running a local website from Synology - is it possible and what are the basics?

I have something I've been building on my hosted website, but I want to move it to my local computer to make a few things easier, namely, to avoid storage issues on the hosted site, and to make it easier for my code to pull assets from local folders where I store things. Long story short, it's basically a collage generator that randomly pulls in images located in about six different folders. I want to avoid the hassle of uploading them to my live site.

Can you run a site locally in Web Station and keep it off the web completely? Also, if it's run locally, but I log into my NAS from somewhere other than my home, am I able to run that site in my QuickConnect session?