Can I break up with someone through text? How bad would that be?

Well, it's not ideal. But since your significant whatever is refusing to talk to you, I don't think it will do any actual harm to your relationship status.

You should try your hardest to talk with him in person, especially because you were together for so long, but if he won't speak and you can't get ahold of him, texting seems like the only other solution

It's done all the time. It's still a cheesy, low down and chicken s**t thing to do.