The Orthodox belief that Mary not only lived in the Temple in Jerusalem, but in the Holy of Holies itself, is commonly attributed to the second century text The Protoevangelium of James. It is known that this apocryphal text is early, because Origen (185-253 AD) refers to some version of it.
This text teaches that Mary was brought to the Second Temple in Jerusalem by her parents at age 3, and then lived inside the Temple - specifically in the Holy of Holies - and was fed by angels, until she was 12 years old:
Protoevangelium of James 8:2-3
2 And when she became twelve years old, there was a council of the priests, saying, "Look, Mary has been in the Temple of the Lord twelve years. What should we do about her so that she won't pollute the sanctuary of the Lord our God?" And they said to the chief priest, "You stand at the altar of the Lord. Go in and pray about her, and if the Lord God reveals anything to you, we'll do it."
3 And the chief priest went in, taking the robe with twelve bells into the Holy of Holies, and prayed about her. And look! An angel of the Lord stood nearby, saying, "Zechariah, Zechariah, go out and assemble the widowers of the people, and let them each bear a staff. And whomever the Lord God points out with a sign, she'll be his wife."
Protoevangelium of James 13:2
2 And Joseph stood from the sackcloth and called her and said to her, "God cared for you. Why have you done this? You've forgotten the Lord your God. Why have you humiliated your soul? You were nourished in the Holy of Holies and received food from the hand of an angel!"
Protoevangelium of James 15:3
3 And the chief priest said to her, "Mary, why have you done this? Why have you humiliated your soul and forgotten the Lord your God? You were raised in the Holy of Holies, and received food from the hand of an angel, and you heard its hymns and danced before it. What is this that you've done?"
The Protoevangelium of James is not regarded as orthodox or canonical by the Orthodox Church, and so citing the above is not an attempt to ascribe this text to them. Rather, it is only mentioned here because it is probably the source of what they believe about Mary, which will be established in the next section from canonical Orthodox sources.
The Entry of the Most Holy Theotokos into the Temple is celebrated by the Orthodox on November 21 every year, as one of the Twelve Great Feasts in the Orthodox liturgical calendar.
Part of official Orthodox liturgy includes a reference to her living in the Holy of Holies:
The Divine Liturgies Of Our Holy Fathers John Chrysostom And Basil The Great - November 21, The Entrance of the Theotokos and Ever-Virgin Mary into the Temple
The angels were struck with amazement beholding the entrance of the Most Pure; seeing how the Virgin entered into the Holy of Holies.
Let no uninitiated hand touch the living Ark of God; but let the faithful lips, singing without ceasing the words of the angel to the Theotokos, cry aloud in great joy: O pure Virgin, you are truly exalted over all.
The Service Books of the Orthodox Church, St. Tikhon's Seminary Press - 21 November, Entrance of the Most-Holy Theotokos
9th Ode of the Canon (Replaces "It is Truly Meet")
The angels beheld the entrance of the Pure One and were amazed! How has the Virgin entered into the Holy of Holies?
As thou art a living temple of God, let no impure hand touch thee, O Theotokos! But let the lips of all believers sing, constantly magnifying thee in joy with the angelic salutation: Truly thou art above all creatures, O Pure One!
This is significant, because unlike the Roman Catholic Church, which is fond of more organized confessions of what they believe, the Orthodox will point those who would learn about their beliefs to their liturgy. A belief making its way into Orthodox liturgy is a canonical endorsement of that belief, by the Orthodox Church.
Further, St. Gregory Palamas (1296-1357 AD), probably the most influential Orthodox Saint of the 2nd millennium AD, wrote on the Feast, and supported the doctrine that Mary lived in the Holy of Holies, and was fed by angels:
St. Gregory Palamas - Sermon on the Entry of the Mother of God into the Temple
Now He finds a Handmaiden perfectly suited to these needs, the supplier of Her own unsullied nature, the Ever-Virgin now hymned by us, and Whose miraculous Entrance into the Temple, into the Holy of Holies, we now celebrate.
In Her manner She showed that She was not so much presented into the Temple, but that She Herself entered into the service of God of her own accord, as if she had wings, striving towards this sacred and divine love. She considered it desirable and fitting that she should enter into the Temple and dwell in the Holy of Holies.
Therefore, the High Priest, seeing that this child, more than anyone else, had divine grace within Her, wished to set Her within the Holy of Holies. He convinced everyone present to welcome this, since God had advanced it and approved it. Through His angel, God assisted the Virgin and sent Her mystical food, with which She was strengthened in nature, while in body She was brought to maturity and was made purer and more exalted than the angels, having the Heavenly spirits as servants. She was led into the Holy of Holies not just once, but was accepted by God to dwell there with Him during Her youth, so that through Her, the Heavenly Abodes might be opened and given for an eternal habitation to those who believe in Her miraculous birthgiving.
The above establishes, at the very least, that this is not a fringe belief in the Orthodox Church.
In the Bible, the Holy of Holies was the place in the Tabernacle/Temple that housed the Ark of the Covenant (Exodus 26:33-34). The High Priest would make an offering there once per year, on the Day of Atonement (Exodus 30:10, Leviticus 16):
Hebrews 9:3-7
3 And after the second veil, the tabernacle which is called the Holiest of all;
4 Which had the golden censer, and the ark of the covenant overlaid round about with gold, wherein was the golden pot that had manna, and Aaron's rod that budded, and the tables of the covenant;
5 And over it the cherubims of glory shadowing the mercyseat; of which we cannot now speak particularly.
6 Now when these things were thus ordained, the priests went always into the first tabernacle, accomplishing the service of God.
7 But into the second went the high priest alone once every year, not without blood, which he offered for himself, and for the errors of the people:
The Holy of Holies in the Temple was not a place which the average Israelite could simply enter, or "tour". It was the most consecrated place in the entire nation. The High Priest alone came in once per year, and when he did, he had blood to perform a blood atonement.
This was a location which God warned even the High Priest not to enter flippantly, or else he would die:
Leviticus 16:2-3
2 And the LORD said unto Moses, Speak unto Aaron thy brother, that he come not at all times into the holy place within the vail before the mercy seat, which is upon the ark; that he die not: for I will appear in the cloud upon the mercy seat.
3 Thus shall Aaron come into the holy place: with a young bullock for a sin offering, and a ram for a burnt offering.
In light of the above, the idea that the High Priest would let (or be allowed to let) a 3 year-old Mary live inside the Holy of Holies, is a joke. It is laughable, both historically, and Biblically, that anyone would ever be allowed to live inside the Most Holy Place.
Further, we would expect to see some trace of Mary's miraculous, sanctified childhood evidenced somewhere in the Gospel accounts of her. Instead, we see things like:
Luke 1:28-29
28 And the angel came in unto her, and said, Hail, thou that art highly favoured, the Lord is with thee: blessed art thou among women.
29 And when she saw him, she was troubled at his saying, and cast in her mind what manner of salutation this should be.
When the angel greets Mary with "Hail, thou that art highly favoured, the Lord is with thee: blessed art thou among women", she is confused, and wonders why the angel greets her in the way that he does. This is not the kind of reaction we would expect from someone who had been raised in the Holy of Holies, and fed with food from Heaven, by angels.
Further, we see no evidence from those who were familiar with Jesus's family that they thought Mary was in any way special:
Matthew 13:54-57
54 And when he was come into his own country, he taught them in their synagogue, insomuch that they were astonished, and said, Whence hath this man this wisdom, and these mighty works?
55 Is not this the carpenter's son? is not his mother called Mary? and his brethren, James, and Joses, and Simon, and Judas?
56 And his sisters, are they not all with us? Whence then hath this man all these things?
57 And they were offended in him. But Jesus said unto them, A prophet is not without honour, save in his own country, and in his own house.
Mark 6:2-4
2 And when the sabbath day was come, he began to teach in the synagogue: and many hearing him were astonished, saying, From whence hath this man these things? and what wisdom is this which is given unto him, that even such mighty works are wrought by his hands?
3 Is not this the carpenter, the son of Mary, the brother of James, and Joses, and of Juda, and Simon? and are not his sisters here with us? And they were offended at him.
4 But Jesus said unto them, A prophet is not without honour, but in his own country, and among his own kin, and in his own house.
Notice what is happening above. Those familiar with Christ's family use the ordinariness of his mother and father as an alibi to reject Him. This is not what we would expect to see, if Mary had been a consecrated baby, allowed to live in the Temple in Jerusalem for many years, under the care of the High Priest, and ministry of the angels.
The belief that Mary was somehow consecrated from birth, or lived in the Temple in Jerusalem, is an example of a belief that has no Biblical or historical support, but rather had its origin in heretical, forged Scripture, and was accepted into sects of Christianity, like the Eastern Orthodox Church, that had misplaced emphasis and fixation on Mary, virginity, and asceticism.