Good day to ask and answer this question, being the Feast of the Nativity of Mary (9 months after the Solemnity of the Immaculate Conception if you're counting)
You know the Bible doesn't have this info in it from the other answers and 2 minutes of poking around. The Bible (in Isiaiah) does prophecy that the Virgin (or at the very least a young woman) will bear a Son, so Mary's not old.
The Apocrypal Gospel of the Nativity of Mary says:
And so she [Mary] reached her fourteenth year; and not only were the wicked unable to charge her with anything worthy of reproach, but all the good, who knew her life and conversation, judged her to be worthy of admiration. Then the high priest publicly announced that the virgins who were publicly settled in the temple, and had reached this time of life, should return home and get married, according to the custom of the nation and the ripeness of their years.
and of Joseph
Now there was among the rest Joseph, of the house and family of David, a man of great age:
which isn't very specific, but I'd imagine it's over 50 at the youngest.
Now, I don't like to think of Joseph as an old man, but it's an old tradition and can't be readily discounted. It's not necessary that James and Jude be half-foster-brothers of Jesus. They could also have been sons of Clophas
Near the cross of Jesus stood his mother, his mother's sister, Mary the wife of Clopas, and Mary Magdalene.
who depending on the interpretation of the commas is also named Mary or not named. I can't tell which, I always thought there were three women, each named Mary at the Cross, but actually reading the Bible is always very surprising. I thought there was also a Salome there. But that's what I read in "Joseph: the man closest to Christ" which is a good book for diving in to all the conflicting traditions concerning young parent Joseph and old parent cherry tree pickin Joseph.
Still, it stands to reason that St. Joseph wasn't that old when he married Our Lady because he brought Jesus up as a "carpenter's Son". He wasn't a long dead obscurity at the time of Our Lord's preaching. The Gospels don't specifically mention that Jesus did carprentry, but that he did practice His father's craft is one of those pseudo-traditional bible inferences people like to make. But even so, carpentry back then (according to Steve Ray in an interview I heard him give more than once on EWTN and Relevant Radio) was more like Stone Masonry than woodwork nowadays. It would have been really, really hard labor and Joseph and Jesus would have had to walk for miles and miles from Nazareth to get to where their work is. And, the reason I'm writing this is just because if, he did work with Jesus, when Jesus was of age (say 14-15, I think), then St. Joseph ,although he was the best man to swing a hammer in human history, probably would have had a rough time doing it into his 60's.
So, best guess. Our Lady was 14-15, her most chaste spouse was 18-25.