Apostolic succession does not mean quite the same thing in Rome and EO, but it does mean for both that the current church is the true successor to the apostles, both as to ordinary and extraordinary offices. The apostles had no extraordinary successors, however, only those in the ordinary offices (as seen in Paul's epistles) of episkopoi, presbuteroi, and diakonoi. Rome thinks that the magisterium particularly and the priesthood generally embody this succession.
The doctrine means then that an unbroken succession can be traced back, on the RCC side, to Peter. No priest in secular order (and certainly not in regular orders) can actually trace back such an unbroken succession. We simply do not have the records to do so. No Protestants can either, of course. In my own case, I had hands laid on me by, among others, Ferguson, Gaffin, and Frame, who I know had hands laid on them that go back to Machen, who had hands laid on him by Warfield, then back through American Presbyterian History and that of the Church of Scotland beforehand.
But none of that is the point. We are the true and proper ordinary spiritual successors of the apostles because we have their doctrine (one of the four attributes of the church, of course, is apostolicity). The business about tracing back has some truth in it as I just noted above--many of us can quickly trace our ordinations back through the laying on of hands--but is not something that we wish to claim since, in fact, neither the RCC nor the EO can establish such in terms of actual historical lineage going back to Christ and his apostles. No church can properly claim this "unbroken lineage" as some establishable historical fact.
Peace,
Alan
Your point echoes Calvin's as Rome puffed out its chest and asked for proof of Apostolicity.
I think there is some importance to the idea that Luther and Calvin were part of the Church catholic when they were Reforming it and didn't simply anoint themselves in a river and declare themselves Apostles.
Many posts have alluded to modern culture. Carl Trueman calls our modern times plastic and notes how pre-modern times had a sense of permanence and fixedness. Life was hard, but one thing pre-modern people didn't struggle with was the question of having to figure out their place in the world. A young boy, growing up in the home of a mason, would be a mason and he might work on a cathedral that his grandfather started and die before its completion.
I have a pretty nice house and even built a steel construction barn, but I don't think it will be standing in 200 years.
I think the idea of an EO Church that feels untouched by modernity is appealing. It's appealing in the way that you go to a place in Europe or Asia and are in a structure that is over 1000 years old.
I grew up Roman Catholic and realize that claims for that communion to have an unbroken "feel" were challenged with Vatican II and shattered with the current Pope.
It's almost better, for some, that they can't even understand what is being said or done because they are participating in (notionally) something unbroken. It's the reason why some Roman Catholics go back to the Latin Mass. They can just pray the Rosary, as generations before, and look up when the bells are rung to see the Priest raise the host at the proper time.
With all of that, however, it's not the permanence of the building or a grandiose, permanent architecture that Reformed worshipers are seeking with respect to permanence. It's not even a certain formula or fixed liturgy. I'll add, as an aside, that even Anglican Churches in England were very modest until less than 200 years ago so it's not even "historically" Anglicn for them to expect extravagance.
The point I make to the Missional/Culturalist wing of my denomination is that I hope that my children's children will be confessing the same Christian truths I am. Even the most ancient buildings change and liturgies may vary, but is Christ and Him crucified, risen, and ascended (and all the evangelical graces) enlivening the hearts and minds of the Church? This is what animates me. I appreciate history and even buildings built to last, but that which I seek to preserve has greater permanence. It goes back to a Promise, made in a garden, before any stone buildings existed.