Totally feasible. I was largely self-taught, though I did begin with one "real" computer class to teach me the principles of coding, object oriented design, etc. I used this class to code for a scientific project producing data for my academic degree (which I ended up never using.) I also did some tinkering around building and overclocking my own computer, installing various flavors of Linux and taking them off again, and acquiring some proficiency with system administration and Unix. Good times.
Thereafter, I learned by biting off more than I could chew; i.e. beginning projects that I lacked the exact skills for, figuring out what I needed to learn, and then learning it by looking up similar code and adapting it to my ends. I started out by volunteering to maintain the website for my son's preschool; once I had access to the code, I decided I was tired of sending out individual emails to parent helpers and reprogrammed the website to send automatic emails to helpers on the calendar. I learned a lot that way. Stack Exchange was a good friend.
Subsequently, I did volunteer work on some other school websites, and parlayed that into a professional job as a web developer and database administrator. I never really struck it big because I could only work from home part time, but it was a rewarding side hustle that kept our noses over water while my husband finished his training, and kept me out of trouble.
Good luck!
[edit to add] I learned the most by starting out with a full website, written by someone else, that needed maintenance and small adjustments. By tinkering with a copy of the website, it was easy to see how changes affected the whole, how different people have different styles of coding, etc. Of course some people are sloppy and less skilled... I remember working on a site written by a guy who had no idea how to use functions, so he would cut and paste the same piece of code a zillion times.
If you are afraid to mess up somebody else's website, what about working on your own? Start out with a canned site like Wordpress, and start messing with customizing the theme or some plugins.
[edit #2] If you're interested in customizing a Wordpress site, check out "Learning PHP, MySQL & JavaScript" by O'Reilly. I like their format.