The Bible does eliminate the possibility. That's the issue. The gospel concerns this life, not the next (although it leads to the next) - it's not a "ticket to heaven" or "get out of Hell free card". It is repentance (a turning from) the flesh (from ourselves) and belief in Christ. It is the Kingdom has come (here on earth) and those who believe in His name (again, necessitating this life) will have eternal life. This is impossible once one has physically died.
There is a reason the Logos became flesh. There is a reason high priests were chosen from among men. There is a reason Christ had to be tempted in all points as we are. When we start giving the false (and unbiblical) hope of "another chance" after death then we are offering another gospel (a different repentance with a different hope).
There are various views on this topic that are Christian. Universal salvation is not one of them. I truly wish it were, as there are friends I've lost who will never again have the chance to hear and respond to the gospel I failed to tell them. But universal salvation is simply a false doctrine, foreign to the Christian faith.
