Most Bibles translate these words as ‘the Most Holy Place’ (which is a part of the temple which housed the Chest of Proofs, commonly known as the Ark of the Covenant). However, the 2001 Translation instead says ‘the holiest of the holy’:
‘Seventy periods of seven Have been set for you and your people And upon the Holy City on Zion,
To bring an end to its sins,
To set a seal on its sins,
To wipe away all their lawless deeds,
And to atone for their errors;
To restore righteous ways through the ages,
To put a seal on the prophecies and visions,
And to anoint the holiest of the holy.’
In the Greek Septuagint, the words in bold read, ‘kai tou chrisai agion agion,’ or, ‘and of anointing the/holy holiest.’ So you can see why some translators would assume this is referring to the Most Holy compartment of the temple. However, this may not be correct.
First, while it’s true that the ancient temple and its ‘Most Holy’ were regularly ‘cleansed’ by the priests (by being sprinkled with water and blood), there is no special concept of anointing the Most Holy compartment.
Second, there is no record of this happening after this prophecy was made. On the contrary, the Most Holy compartment was destroyed in 70 CE along with the rest of the second temple.
Third, since this prophecy is widely believed to be about Jesus, it could be referring to him. Jesus was, of course, anointed as King.
Of course, it could be that these words are indeed referring to the Most Holy compartment, but in some kind of symbolic way.
Therefore, since there is this uncertainty, this translation simply translates the Greek as literally as possible, saying ‘to anoint the holiest of the holy’. You, the reader, can decide what it’s talking about.