So, you have a lot of questions in a lot of areas. 1st, HTML 5 is just the version of a language. Your server can interpret and vend files written in this version of HTML provided you code correctly. So the technical part is yes, you can host a site from your server.
But then you get into the practicality issues, such as "does my computer need to be on for people to access it." Obviously, the answer is yes. If you are hosting a web site, the rules of thumb are:
a: turn on the server
b: leave the server running
c: have a statically assigned IP address assigned to the server
d: properly configure DNS so users can reach the site
e: allow search engines to troll your site (or control access using a robots.txt file)
Now, beyond these steps, you should also consider why a hosting company is charging you to host the site. The simple answer is that they are paying for the things that you do not want to... Redundancy. The hosting provider will have redundant Internet connections, redundant power, cooling, servers, and likely geographically distant redundant locations. This means that if one facility fails, your site remains available. When you self host, you are subject to many single points of failure, such as power, Internet, the server itself, Mayan end of days, fiscal cliffs, and tribbles.
So, to sum up. Yes, OS X can do that. Should it? That is a question you must answer based on your risk assessment.