Luke 1:5-7

During the reign of Herod king of Judea, there lived a priest named Zechariah who belonged to the priestly division of Abijah, and he had a wife named Elizabeth, who was a descendant of Aaron. They were both righteous in the sight of God, following all the commandments and ordinances of the Lord blamelessly. But they did not have a child, because Elizabeth was barren, and they were both very old.
Luke 1:5-7, New English Translation (NET)
 
We turn to Adam Clarke and his Commentary on the Bible to provide with background information on Zechariah and Elizabeth:
 
In the days of Herod, the king - This was Herod, surnamed the Great, the son of Antipater, an Idumean by birth, who had professed himself a proselyte to the Jewish religion, but regarded no religion, farther than it promoted his secular interests and ambition. Thus, for the first time, the throne of Judah was filled by a person not of Jewish extraction, who had been forced upon the people by the Roman government. Hence it appears plain that the prophecy of Jacob, Gen_49:10, was now fulfilled; for the scepter had departed from Judah: and now was the time, according to another prophecy, to look for the governor from Bethlehem, who should rule and feed the people of Israel: Mic_5:1, Mic_5:2. See a large account of the family of the Herods, in the note on Mat_2:1 (note). This was before Christ six years.
 
The course of Abiah - When the sacerdotal families grew very numerous, so that all could not officiate together at the tabernacle, David divided them into twenty-four classes, that they might minister by turns, 1Ch_24:1, etc., each family serving a whole week, 2Ki_11:7; 2Ch_23:8. Abiah was the eighth in the order in which they had been originally established: 1Ch_24:10. These dates and persons are particularly mentioned as a full confirmation of the truth of the facts themselves; because any person, at the time this Gospel was written, might have satisfied himself by applying to the family of John the Baptist, the family of our Lord, or the surrounding neighbors. What a full proof of the Gospel history! It was published immediately after the time in which these facts took place; and among the very people, thousands of whom had been eye-witnesses of them; and among those, too, whose essential interest it was to have discredited them if they could; and yet, in all that age, in which only they could have been contradicted with advantage, no man ever arose to call them in question! What an absolute proof was this that the thing was impossible; and that the truth of the Gospel history was acknowledged by all who paid any attention to the evidences it produced!
 
Of the daughters of Aaron - That is, she was of one of the sacerdotal families. This shows that John was most nobly descended: his father was a priest and his mother the daughter of a priest; and thus, both by father and mother, he descended from the family of Amram, of whom came Moses, Aaron, and Miriam, the most illustrious characters in the whole Jewish history.
 
They were both righteous - Upright and holy in all their outward conduct in civil life.
 
Before God - Possessing the spirit of the religion they professed; exercising themselves constantly in the presence of their Maker, whose eye, they knew, was upon all their conduct, and who examined all their motives.
 
Walking in all the commandments and ordinances of the Lord blameless - None being able to lay any evil to their charge. They were as exemplary and conscientious in the discharge of their religious duties as they were in the discharge of the offices of civil life. What a sacred pair! they made their duty to God, to their neighbor, and to themselves, walk constantly hand in hand. See the note on Mat_3:15. Perhaps εντολαι, commandments, may here mean the decalogue; and δικαιωματα, ordinances, the ceremonial and judicial laws which were delivered after the decalogue: as all the precepts delivered from Exo_21:1 to Exo_24:1 are termed δικαιωματα, judgments or ordinances.
 
Both were now well stricken in years - By the order of God, sterility and old age both met in the person of Elisabeth, to render the birth of a son (humanly speaking) impossible. This was an exact parallel to the case of Sarah and Abraham, Gen_11:30; Gen_17:17. Christ must (by the miraculous power of God) be born of a virgin: whatever was connected with, or referred to, his incarnation must be miraculous and impressive. Isaac was his grand type, and therefore must be born miraculously - contrary to the common course and rule of nature: Abraham was a hundred years of age, Sarah was ninety, Gen_17:17, and it had Ceased to be with Sarah After The Manner Of Women, Gen_18:11, and therefore, from her age and state, the birth of a child must, according to nature, have been impossible; and it was thus; that it might be miraculous. John the Baptist was to be the forerunner of Christ; his birth, like that of Isaac, must be miraculous, because, like the other, it was to be a representation of the birth of Christ; therefore his parents were both far advanced in years, and besides, Elisabeth was naturally barren. The birth of these three extraordinary persons was announced nearly in the same way. God himself foretells the birth of Isaac, Gen_17:16. The angel of the Lord announces the birth of John the Baptist, Luk_1:13; and six months after, the angel Gabriel, the same angel, proclaims to Mary the birth of Christ! Man is naturally an inconsiderate and incredulous creature: he must have extraordinary things to arrest and fix his attention; and he requires well-attested miracles from God, to bespeak and confirm his faith. Every person who has properly considered the nature of man must see that the whole of natural religion, so termed, is little else than a disbelief of all religion.