Sure you can - I did - https://ignitedds.com
We've been around for 4 years now - have thousands of students and I'm still the only developer on it. It was originally coded in what you listed - HTML / CSS / Javascript / PHP / MySQL. About a year and a half ago I re-wrote the entire thing to use AngularJS / Firebase. I still keep PHP / MySQL around for a handful of tasks (reporting, etc...)
It's an incredible amount of work to do it on your own (while maintaining a day job) but it's definitely possible. You just need to have focus and determination! (to say the least hah)