Small cities or training centers with guanxi's "might" be able to swing you with a TEFL, it's a little harder these days, it's hit or miss.. But I know most bigger cities won't accept it because there is enough BA/BS/MA and PhD talent coming through the legal channel these days.

I won't lie, it's going to be tough.

Yes mate, you can.  China has hundreds of cities that have over a million people in them.  Look to the road less travelled.  

But, choose wisely.

Since the last revision of the visa law (in September 2013) it has become almost obligatory in the most developed cities to have at least a Bachelor's, a 20h TEFL (or the equivalent for other languages) and 2 years of related experience that can be proven (with letter of recommendation from your ex employers for e.g.). Less developed and smaller cities are less regarding.

Also the government is cracking down on foreigners working illegally in China right now, 300 foreigners got deported a week ago, 200 a month ago, ..., do not accept to work on a F or a L visa, do not trust any employer who is telling you that you can work on anything else than a Z visa, do not trust employers claiming that they have good relations and that you won't get into troubles, if they would have such good relations they would be authorized to get Z visas, you can take the risk to work illegally of course but don't come to cry if you get yourself into troubles.

additionally you need the relevant degrees now...