“they both were advanced in their days.” The Levites could only work from 20 to 50, but the priests could work as long as they were able. There is a very good chance that both Elizabeth and Zechariah were over 60. It is almost certain that they both died before John started his ministry.
The times of service differed for the priests and Levites. When the Levites first started their service under Moses, they were counted for their duty from age 30 to 50, a period of 20 years (Numbers 4:3, 23, 30, 35, 39, 43 and 4:47). However, Numbers 8:24-26, also written during the wilderness wanderings, says the Levites served from age 25-50. Although it is possible that the number 20 was very temporary and revised up to 25, it is more likely that the Levites started an apprenticeship before they took over the full responsibility of their duties. King David revised the ages, and the Levites started ministering at age 20 (1 Chron. 23:24-26; cp. 2 Chron. 32:17; Ezra 3:8).
When it came to the priests, “There was not any fixed age for entering on the office of the high-priest, any more than on that of an ordinary priest” (Alfred Edersheim, The Temple: It’s Ministry and Services as They Were at the Time of Christ, Wm. B. Eerdmans, Grand Rapids, reprinted 1978; p. 94). The High Priest, for example, started when the High Priest before him died, no matter how old he was, and ministered as High Priest until he died. The Law never specified when the priests started their office, but when the Talmud was written, it said age 20.
“barren.” In a culture in which children were considered the blessing of the Lord, and the death rate was so high that each couple had to have 5 children to keep the population number stable, being barren was considered a curse. In fact, the situation highlights the character of Zechariah, who was no doubt under pressure to divorce her. There were people who considered it a religious duty to divorce a barren wife (Edersheim, Life and Times, book II, p. 137). No wonder Elizabeth said she had “disgrace among people” (Luke 1:25).