What does Acts 2:24 mean?

This is one a significant truth in the story of salvation and a pivotal aspect of the nature of Jesus. "Not possible" is from the Greek root word ou which is a simple negative. Jesus remaining dead was not to be; it could not happen. And so God raised Him. God was the appeals judge, overriding the conviction and punishment of the human courts. Jesus was wrongly accused (Mark 14:55–59), wrongly convicted (John 19:12–16), and wrongly executed (Matthew 27:54). Because He is perfectly sinless, it was not possible that He would stay dead.

This statement in this time and place is also one of the strongest answers to the claim that the disciples faked the resurrection. Peter is preaching in Jerusalem about fifty days after Jesus' resurrection. Paul later says that Jesus "appeared to Cephas [Peter], then to the twelve. Then he appeared to more than five hundred brothers at one time, most of whom are still alive, though some have fallen asleep. Then he appeared to James, then to all the apostles" (1 Corinthians 15:5–7). Even the Sanhedrin and the guards know there is no body (Matthew 28:11–15). No one in Jerusalem has evidence to contradict the apostles' claim that Jesus has risen—though some attempt to spread rumors anyway (Matthew 28:11–15).

Now, Peter turns to David to show that Jesus of Nazareth is the Messiah of the Jews and that King David prophesied His resurrection hundreds of years prior. It's significant that an executed man came back to life; it's stunning that the Messiah was in their midst and they missed it.