How are the sinners to view the hope of eternal life in heaven? Is there room to assume they can obtain it?
First of all, sinners are a "we" term and never a "they". We are all in the mix; presidents and preachers, powerful and poor, politicians and paupers....we are them. No one is exempt from such a label.
But rather than living a life of defeat we each have great reason to anticipate perfect/complete hope. We are good enough in God's eyes because Jesus died on the cross. The cross isn't a Sunday School form from which the deacons have ample supply of Sunday topics. It is the highest of heaven's technology to help mankind start life over.
Who wouldn't love to start life over? Who wouldn't wish for the past "bads" to go away? Who wouldn't like to pass jail and move on to "Go"? The cross of Jesus is first of all historic and then it is mankind's launching pad into profound terrain. Nobody can hardly believe we can be born over...really....as Nicodemas properly suspicioned in John 3. Yet, Jesus said it is precisely possible.
Do the sinners have a legitimate chance at being spared by the Powerful God? Absolutely. My advice would be to accept the work He did on the cross rather than doubt it. God isn't looking for theologians. He is looking for any too impoverished in spirit to save themselves. He would love to do that for us.
Our job, therefore, is to let Him. That's why they call Him the savior and not us.