Paul says depending on the law in order to get to heaven is a curse. One just will not make it by depending upon the law because humans fail way to much in comparison to the demands of a Holy God.
But just for arguments sake, what is covering your sins then if one believes Christianity and Paul is wrong?
Paul is wrong. That is basically saying that G-d is tricking us or a liar (G-d forbid) G-d says in Deut 30, regarding the law. (paraphrasing)...This law is not to difficult for you....it is not out of reach... is it up in heaven that it must be brought down? Is it across the sea that we must go and get it? NO! It is near to you!...in your mouth and in your heart that you may do it! Was G-d just saying that? Knowing that it was really a curse? He let his chosen people suffer under the illusion that the law was sufficent, when it really wasnt for 1,500 years from Moses to Jesus? G-d forbid!
The "old testament" people did not get to heaven by doing the law only anyway. Forgiveness (many examples in the OT) is through prayer, repentance, fasting and charity. That coupled with doing ones best to do-keep the mitzvot (commandments)is what gives us a relationship with G-d and forgiveness.
According to Pauls view (or perhaps the church) David would not have gone to heaven. There is not a single case in all of scripture where someone sins and is then going off to do a blood sacrifice for forgiveness! Why? A)Because it is the LEAST effective means of forgiveness (prayer, repentance and obedience is superior) and B) Sacrificial sins were primarily for UNintentional sins. Only about 6 intentional sins could be atoned for by sacrifice (and even then blood is not required, Lev 5:11,12) The rest of intentional sin has ALWAYS been atoned by G-ds grace through prayer, repentance, fasting and charity.
And that is the answer to your question...