This is all personal opinion, obviously, but:
I don't think it is literally impossible to get a job teaching English without a degree, but I think it will be extremely difficult. Apart from JET, most of the major eikaiwa and organisations like Interac (ie. contracting organisations that send teachers to schools) currently require a degree, and while there are smaller language schools, there are lots more people looking for jobs than there are positions to fill. If you're not already in Japan, you'll also need to find someone to sponsor your visa, and if you look through the job listings you'll see that not many of them are willing to do that (I guess because there are plenty of candidates who already have a visa).
The other thing is that teaching English is not an ideal way to learn Japanese. A lot of English-teaching positions require you to work pretty hard, so you'll be using English all day. I'm currently on the JET programme and I'd say I'm on the cusp of passing JLPT2, but while my understanding has got better, my spoken Japanese has got much, much worse since getting here because I spend all day teaching, speaking, and correcting in English, and when I get home my wife refuses to speak to me in Japanese because she wants to maintain her English. (And when I try to speak to her in Japanese we always end up rowing! Maybe time to find a conversation partner on Skype!)
So my advice would be to keep looking through all the job listing sites in case you can find teaching positions that don't require a degree. If you're really interested in teaching English then yes, by all means pursue some sort of English-teaching qualification because it can only improve your chances. And if you're really interested in coming to Japan, then try to find out what other opportunities might be available and continue to work hard at improving your Japanese.
Good luck.
PS. You might find some interesting links
here.