Far from being completely impossible, subjective ethics are entirely more consistent than the alternative. The important thing about analyzing ethical systems is to find axioms that you actually believe. An unanalyzed moral system tends to have false axioms: things that the speaker says are fundamental tenets, but on further analysis simply don't stand muster.

For example, the Commandment "Thou shalt not kill" is anything but an axiom. All but the most unreasonable followers will be able to produce some circumstance where killing is moral. That's not an axiom, that's a guideline.

So why find one's axioms? Ultimately, the entire study of ethics boils down to one simple question: "What ought I do?". When faced with a dilemma, what should I choose? With the understanding of your beliefs, you can make a reasoned decision, consistent with your broader choices. If you exist in a society, then you will inevitably be forced to react to the actions of others, which, in of itself, is a forced action on your part. If you don't have some framework with which to judge the actions of others, then your ethical framework is utterly incomplete.