Fr Ambrose, I have written an earlier post on this subject that sets out what we know about who will be saved and who will not be, but I'll repeat the details here.
Firstly, Mark 16:16: “He that believeth and is baptized shall be saved; but he that believeth not shall be damned”. This passage proves that you have to know the true faith to be saved.
Secondly, John 3:5: "Verily, verily, I say unto thee, Unless a man is born again of water and the Spirit, he cannot enter into the Kingdom of God”. This passage proves that you have to receive true baptism to be saved. So you need BOTH Baptism AND the True Faith to be saved.
Thirdly, John 6:53: “Verily, verily, I say unto you, Unless ye eat the Flesh of the Son of Man, and drink His Blood, ye have no life in you”. This passage proves that apart from true doctrine and Baptism, you need the Divine Eucharist to be saved.
True faith, of course, is the Orthodox faith. All dogmatic deviations from the Orthodox faith are, as I'm sure you would agree, incompatible with this necessary criterion of salvation. What is Orthodox and what is not can be known from the Creed and from the dogmatic decrees of the Church Councils and Fathers.
We also know that true Baptism and the true Eucharist do not exist outside the Orthodox Church. Thus, the 46th Apostolic Canon:“We order that a bishop or presbyter that recognized the baptism or sacrifice of heretics be defrocked. For ‘what accord has Christ with Belial? Or what has a believer in common with an unbeliever?’” And the second canon of St Cyprian of Carthage: “We declare that no one can possibly be baptized outside the Catholic (i.e. the Orthodox) Church, there being but one baptism, and this existing only in the Catholic Church.”
Being saved in this context means being delivered from Hades after death. We do not know that everyone condemned to Hades will also be condemned to Gehenna, i.e. the eternal fire, at the Last Judgment, because we have testimonies of some souls being saved from Hades by the prayers of the Church. From this evidence we know that condemnation to Hades is not necessarily eternal and hence we can hope that God will have mercy on the souls of unbelievers at the final judgment.
But that those outside the Church will certainly go to Hades after death is proved by the following passage from the vision of Blessed Theodora:
"Those who believe in the Holy Trinity and take as frequently as possible the Holy Communion of the Holy Mysteries of Christ, our Saviour's body and Blood - such people can rise to heaven directly, with no hindrances, and the holy angels defend them, and the holy saints of God pray for their salvation, since they have lived righteously. No one, however, takes care of wicked and depraved heretics, who do nothing useful during their lives, and live in disbelief and heresy. The angels can say nothing in their defence... [Only those] enlightened by the faith and holy baptism can rise and be tested in the stations of torment [that is, the toll-houses]. The unbelievers do not come here. Their souls belong to hell even before they part from their bodies. When they die, the devils take their souls with no need to test them. Such souls are their proper prey, and they take them down to the abyss."
As to the likelihood that those condemned to Hades for lack of true faith and baptism will also be condemned to Gehenna, that depends on whether or not they consciously resisted the truth. Metropolitan Philaret of New York's famous essay "Will the heterodox be saved?" deals with this question in detail. His conclusion was that it will go worse in the end for those who consciously resisted the truth than for those who did not, i.e. who acted out of ignorance. Only God knows the heart of man in this matter.
The other question to consider is the ignorance of unbelievers, and whether it is voluntary or involuntary ignorance. Ignorance can be voluntary if the individual does not take the necessary steps to discover the truth.
Thus, Luke 12:47-8 "That servant who knew his master's will, and prepared not himself, neither did according to his will, shall be beaten with many stripes. But he that knew not, and did things worthy of stripes, shall be beaten with few stripes. For unto whomsoever much is given, of him shall much be required; and he to whom men have committed much, of him they will ask the more".
Blessed Theophylact comments: "Here some will object, saying: 'He who knows the will of his Lord, but does not do it, is deservedly punished. But why is the ignorant punished?' Because when he might have known he did not wish to do so, but was the cause of his own ignorance through sloth."
And St Cyril of Alexandria: "How can he who did not know it be guilty? The reason is, because he did not want to know it, although it was in his power to learn."
And Bishop Nikolai Velimirovich: "Are the people at fault if godless elders and false prophets lead them onto foreign paths? The people are not at fault to as great an extent as their elders and the false prophets, but they are at fault to some extent. For God gave to the people also to know the right path, both through their conscience and through the preaching of the word of God, so that people should not blindly have followed their blind guides, who led them by false paths that alienated them from God and His Laws."
We know there is no excuse for ignorance of One God the Creator. Thus, St Jerome: "Ours and every other race of men knows God naturally. There are no peoples who do not recognise their Creator naturally.”
And St John Chrysostom: "From the beginning God placed the knowledge of Himself in men, but the pagans awarded this knowledge to sticks and stones, doing wrong to the truth to the extent that they were able."
And again: "One way of coming to the knowledge of God is that which is provided by the whole of creation; and another, no less significant, is that which is offered by conscience, the whole of which we have expounded upon at greater length, showing how you have a self-taught knowledge of what is good and what is not so good, and how conscience urges all this upon you from within. Two teachers, then, are given you from the beginning: creation and conscience. Neither of them has a voice to speak out; yet they teach men in silence."
And finally, the story of St Barbara, who was brought up in polytheism, but came to know the One God through the evidence of Creation and her own conscience, and through her prayers to God the true saving doctrines of the Incarnation and the Trinity were revealed.
Therefore, where there can be involuntary ignorance, there is no such thing as invincible ignorance, since the Providence of God overcomes everything, provided that the will of the individual cooperates. As St John Cassian says: "When God sees in us some beginnings of good will, He at once enlightens it, urging it on towards salvation."
This is what the Church teaches about the necessity of true faith and sacraments for salvation. As you can see, I am quite right to insist that no one can be saved outside the Church, provided we understand salvation in this case to mean salvation of the soul immediately after death. As to salvation at the Last Judgment, when we are judged in both soul and body, that is up to God alone. But I'm sure you can agree that those condemned to Hades are much more likely to be condemned to Gehenna at the last day than those who are not.