How is the neutral kaon stable?

The kaons don't consist of two particles which are each other's antiparticle. The kaon comes in four varieties:

1,2) an up-quark with an anti-strange-quark ($ubar{s}$) and vice-versa ($bar{u}s$)
3,4) a down-quark with an anti-strange-quark ($dbar{s}$) and vice-versa ($bar{d}s$)

Since the down-quark and strange-quark have the same electric charge, these are the neutral kaons. You can figure out for yourself that the other two have an electric charge of $+1$ and $-1$.

All these four decay rapidly, but not because the two quarks are each other's anti-particle. For more info see this Wikipedia article (it's always handy to look on Wikipedia first before asking a question).