Not sure if you meant having an HTML/JS app served from GitHub with database capabilities, but maybe you'd want to check out something like StackMob? (Disclaimer, I work at StackMob, but appearing here as a fellow developer)
You can commit your HTML/JS files to GitHub and StackMob will serve those pages for you via the GitHub integration StackMob has. What that enables you to do is actually use StackMob's REST API for its backend services (datastore, versioning, custom server side code) via AJAX since there's no cross-domain AJAX problems (pages and ajax calls are going to/from your stackmob.com hosted pages)
I use it to host my fiancee and I's wedding site and I use the datastore to hold guest information, people can register/login etc. The data is stored on Stackmob via the AJAX calls by using the JS SDK StackMob provides.
Example javascript:
var user = new StackMob.User({ username: 'asdf', password: 'asdf', acquaintance: 'bride', ...});
user.create(); //this fires off the ajax call to StackMob and saves your data in the datastore on that end.
It's free at a certain usage level (I'd imagine your personal use would be within the free threshold) if you wanted to give it a go.
StackMob has that prebuilt backend that you're talking about. It has the REST API and datastore/schema generation built already so that you can just start connecting and saving right away. By the way I do work at StackMob, so in the interest of full disclosure I ought to mention that! But I'm here as a fellow frontend developer.
I had to make a site for my fiancee and I's wedding, and I realized I could whip one up really quick and just stick it on StackMob - backend, hosting, everything.
An example saving/fetching data to/from StackMob via the StackMob JS SDK (built on Backbone.js):
var user = new StackMob.User({ username: 'ericktai', password: 'opensesame', ..});
user.create(); //this saves your user to the datastore on StackMob's servers in the background via AJAX
var users = new StackMob.Users(); //notice plural
users.fetch(); //this fetches all users
var user = new StackMob.User({ username: 'ericktai'}); //notice singular
user.fetch(); //this fetches my user from StackMob
Sounds like you're writing a JS web app, so you can also try our JS SDK that lets you connect to StackMob's backend - it's your datastore already built on servers. Here's a tutorial I wrote up to give you an idea of what that actually means programatically.
To make sure you don't run into any cross domain AJAX call issues, we host your html app also. This relieves you of having to worry about whether browsers support cross-origin headers or not.
Good luck to you developing! It's great that there are so many tools out there nowadays like aespinoza mentioned! Find the tool that works best for you :)
and to the other commenters point, we allow you to write custom server side code as well - accessible via a REST API that you can extend with your code
StackMob provides a platform for mobile apps which includes a backend/datastore that's running for you in the cloud running behind a REST API. Push messaging is also available, user authentication.. The things you build for your backend infrastructure - except that it's already built. Just start using the StackMob SDKs to help you take advantage of the services.
This particular component is part of our HTML5 offering - in particular how you would host your HTML5 files on StackMob via GitHub. We provide a JS SDK that enables your web app to access StackMob (datastore, push, etc). To accomplish that, and to make sure that any browser can support it (and not depend on implementation of CORS - cross origin - headers), we provide hosting for your web apps so that there's no potential of cross-domain rejections. The aim for that is to support as broad an audience for your app as possible - not just the latest browsers that support CORS. It also helps answers security concerns we had, as we didn't want your OAuth keys floating out there in your JS source.
Just a quick snippet of what we do in programmatic terms. The following code via the StackMob JS SDK saves your user object to StackMob servers.
var user = new Stackmob.User({ username: 'Chuck Norris', password: 'my fists', age: 2000, weaponofchoice: 'nunchucks' });
user.create({ success: function(model) { /* saved! */ } } ); //this fires of AJAX that saves your user
We'll need to do a better job of encapsulating that in a digestable format for news consumption, but I do hope that helps a bit!