The most common and most likely answer to this crossword clue is the 11 letter word RADIOCARBON
I've seen this crossword clue in The Guardian
R
A
D
I
O
C
A
R
B
O
N
Buddy explains!
The straight part
We know that "dating agency" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dating agency = radiocarbon.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.