Home-schoolers aren't failures. If you're doing your work, you're doing the same thing that you would be doing in public school, but without all the torment. I, too, left public school towards the end of my high schooling years to be home-schooled because I couldn't deal with all the stuff I had to put up with there all the time. In fact, I did a better job because I didn't have to deal with people picking on me on a regular basis, so I could actually concentrate on achieving something academically, rather than being controlled by those anxieties and fears that I faced every day. I'm sorry that you're being picked on at home as well.
The other thing I will tell you is that your post makes you sound really timid. Unfortunately, people like to pounce on that the moment they see it. I know that you can't just snap out of those anxieties, I have had and still have a lot of them myself, but that's why you need to fake being more confident. Make sure you look everyone in the face and whenever someone insults you, try to insult them back. I don't know why, but most human beings try to test and see how much they can take advantage of the people around them and if it's a lot, they often tend to become really abusive towards that person. So a lot of really nice people get taken advantage of.
I know exactly how this advice sounds to you because I've been in your shoes before, but it'll help you if you can do it. I've gotten better at it and people pick on me less than they used to. And let me tell you, I fake it A WHOLE LOT. Sometimes people start calling me names and inside my heart is pounding and I feel like running and like I'm the biggest loser ever, but then I turn that feeling into false bravado and just yell at them back and they often don't see through me.
It also helps being around older adults more often and trying to befriend them, which I did as well because they are much more mature, in general, than people your age and will probably think your kind nature is refreshing rather than something they need to take advantage of.
But don't give up or let your fears control you. I'm glad that you're here and trying to reach out to people. Because fear or rejection just guarantees you'll be alone. Yea, it's scary to face it, but there's no other way to have people in your life. Good luck with everything. And you're not a loser.