Non-gymnast, not terribly flexible. Willing to spend 1-2 years doing exercises to train flexibility and strength (though hopefully not exceeding 30 minutes a day).
How would one go about this? What drills and interim goals should one set along the way?
Can you do a front handspring? Are you male of female and how old are you? I used to be a pretty accomplished floor gymnast and a back handspring was a trick I could never do. It is dangerous if not done properly and you can literally break your neck trying. The best thing to do is to take some actual gymnastics lessons and learn a bunch of floor tricks and build up to a back handspring.
Your arms need to strong enough to absorb the impact of landing on your hands. Spend some time doing handstands with your back against the wall. Once you can balance fairly well against the wall, start doing handstand push-ups. (Like it sounds. Do a handstand, bend elbows, push back up to handstand.)
You’ll need some flexibility in your back, but perhaps not as much as you might think. Also, it’s upper back flexibility that’s most important in a back handspring - your lower back shouldn’t be bending too much. You want to do stretches that open up your shoulders.
The biggest hurdle in learning a backhandspring is getting over the fear of flinging oneself backwards. Find something soft and big and about waist height. Face away with arms overhead. Sit back slightly. Legs should not be bent more than 90 degrees. Shoulders should be directly above hips. As you sit, your arms swing down to your side. Then, from the sitting position, fling your arms up to your ears and yourself backwards so that you land flat on your back on the big squishy thing. Your arms should be straight and along side your ears. You should not be bent at the waist.
The next drill is to do an actual backhandspring on an incline mat with a spotter. You stand at the top of the mat and do a back handspring down the hill. The spotter provides some extra support.
An alternative next step is to try an actual backhandspring on a trampoline with a spotter. I suppose that trampolines might be easier to find than incline mats outside of a gym.
However, I’d recommend finding a gym and paying for some private lessons or taking some classes. It’s best to do all this with someone who knows what they are doing and can tell you if your form is off. Bending at the hips is probably the most common rookie mistake and will almost always lead to an unpleasant landing on the head.