Rashi understands that Rivkah was three years old when she married Yitzchak, that ten years later, at the age of 13, they tried having children, and ten years after that, at the age of 23, they prayed to Hashem, afraid she was barren, and then she had Esav and Yaakov.
There are various textual inputs into this chronology, yet other accountings are possible, depending on how one understands the pesukim, and these other accountings are offered by other commentators.
In short, for Rashi's accounting:
There are four verses mentioning age. Sarah has Yitzchak when she was 90. Sarah died at 127. Yitzchak married Rivkah at 40. Yitzchak had Yaakov and Esav when he was 60.
Since the Binding of Yitzchak adjoins word coming to Avraham of Rivkah's birth, an assumption is made that her birth occurred immediately after. Similarly, since Sarah's death adjoins the Binding of Yitzchak, an assumption is made that her birth occurred immediately after, according to one midrash as a result of hearing that Yitzchak was to be slaughtered.
Since arranging a marriage for Yitzchak adjoins the death of Sarah, and since Rivkah's presence comforts Yitzhak for the loss of his mother, we may assume that this happened some time after.
Now, the calculations. Since Sarah had Yitzchak at 90 and died as a result of the Binding of Yitzchak, Yitzchak must be 37 at the Binding. Since word of Rivkah's birth is assumed to occur at that point, when Yitzchak is 37, Rivkah is just born.
Assuming Yitzchak marries Rivkah when she arrives, and another verse says he was 40 when he married her, she must be 3 years old at this point, and halachically, kiddushin/nisuin is tofes
at that age.
Halachically, one must worry about barrenness after ten years of trying without results. We see from pesukim that they prayed when Yitzchak was 60. Thus, they were trying for 10 years. Earlier than that, she could not have become pregnant. Thus ends the calculations.
Note that one could challenge this based on questions of whether juxtaposition really proves that Yitzchak was 37 by the Binding, even if word of Rivkah's birth does in fact equal her being born just then. Or, one could question whether the juxtaposition of a genealogical section that Rivkah was born implies that she was actually born at that time. So we need not say that Rivkah was 3. And so on. Certain assumptions and constraints lead to certain conclusions, and other ones may lead to other conclusions.
Certain nuances of certain pesukim can also be taken to imply an older Rivkah.
However. There is a big difference between saying that there are various options, and one is more plausible than another, and stating that a specific option is entirely implausible and obscene to boot. I wish to address both charges in turn.
First, the implausibility. What exactly is implausible? I can point out several potential aspects of the story that seem implausible, at least assuming absence of miracles (which, by the way, a serious midrashist like Rashi would not discount.)
1) Could a three year old carry a pitcher, either empty or full of water?
I have a two year old son, and even before he turned two, he was able to carry, with some difficulty, a 1 gallon Poland Spring bottle across the kitchen floor. He liked doing this. And a three year old can be 3 years and 6 months.
Now, you might not have a baby, to be able to make such assessments. Or perhaps your baby is not doing it, and mine is atypical. But that does not mean it is impossible. And normal is different for different societies. Here is a composite of various children getting water. In particular, the one in the top left corner strikes me as being three or four years old.
Would anyone seriously send a child to fetch water from the well? Could a child hold his/her own?
Now, in New York City, one would be foolish to send a three year old to do this. But this reflects urban Western values of the 20th-21st centuries. In other countries, in rural areas, in more "primitive" societies, this might be perfectly natural.
When in Israel many years ago, I would see a 5 year old boarding the Egged bus taking her younger three year old sibling to school. In a trusting society (such that they are not afraid of kidnapping) where they have certain expectations of responsibility in children, the children might just live up to it.
There was a study a while ago, I think of Malaysian children, in which they found that these children did not go through a crawling stage. It was expected that these children would walk, much sooner than we expect children to walk in the West, and indeed they walked much earlier.
2) Is it logical to expect a three year old to carry on such conversation with Eliezer?
Well, my son is able to carry on conversation, perfectly grammatical (except consistently replacing "I" and "me" with "you"), in well structured paragraphs. And he is two. I'm sure some three year olds can do so as well.
Even if not, this may well be a paraphrase of the conversation. Do you think Pharaoh spoke Hebrew and said exactly the stilted speech attributed to him. It's a summary.
3) Would a three year old display such chessed? Doesn't that take some maturity?
Possibly. Yesterday I put my son in front of Sesame Street, and perhaps the volume was a tad too high. My wife was sleeping in the same room. When I came back, the TV was off. I asked him what had happened, and he told me he had turned it off "so that Mommy could sleep in the morning."
4) Would they really trust a three-year old to decide whether to marry Yitzchak?!
As Shadal points out, the question of whether or not to marry Yitzchak was not put to Rivkah. They already had said that the matter had come from God and that they thus could not divert to the right or left. This was most certainly an arranged marriage. The question they put to her was whether she wished to go with Eliezer immediately, or leave in a year or ten months (as classic Jewish translations go).
I would also point out that she brings her nursemaid, quite appropriate for a three year old. On the other hand, one can simply say (IIRC Shadal does) that the practice was that the nursemaid would act as a governess for years to come.
Isn't this obscene? That is the second consideration. And if it is obscene, then it is also implausible, for Yitzchak is not meant to be portrayed as obscene. Nor would Rivkah's family members agree to such an obscene arrangement.
To which I could answer: No, not really.
First, point out that it would seem that according to Rashi, they did not consummate at 3, but ten years later. Still, she is quite young! And he is 50!
Obscenity is relative to the culture in which you live. Nowadays we would not commend a man who married his niece, but this is a commendable practice in Talmudic times. And we would not say that this practice (of marrying a niece) is implausible as a result, such that one must reinterpret the gemara. Nor necessarily would we call it obscene, given the social mores at the time and place.
We must take be wary about judging obscenity and plausibility based on Western 21st century attitudes. The same folk who scoff at Chazal when they say the Avot kept the Torah, even the Rabbinic commandments, and consider this anachronism, do not think twice at raising objections to actions of the Avot from mores and practices of 21st century New York!
Differences in age did not necessarily matter so much back then, particularly in arranged marriages. And 12/13 was viewed as a perfectly fine time for consummation with one's spouse. Nowadays, we have different views, based on an understanding of emotional maturity and aility to have informed consent. (Maturity might differ or matter based on the society.) And we would consider this pedophilia.
On a related note, where do we see someone enter someone's household but the consummation happen years later? Well, we have Mohammed, who married one of his wives at the age of 6 and consummated the marriage when she was 9. And this is not related as a bad thing. It was accepted practice in his society at that time.
Look to the Torah. Our Sages' understanding of Amah Ivriyah (not practiced nowadays, I should note) is that she is taken as a maidservant into her master's house, and gets out either by developing signs of maturity, or her master or master's son marrying her. And there is some expectation that one of the latter two will take place, and a sense of betrayal if it does not. One could draw parallels to a Rivkah situation, with consummation when she reaches a certain stage of physical maturity.
Or look to kohen gadol and certain interpretations of the requirement of betulah, which I won't elaborate upon here.
In sum, I am a pashtan and favor the alternate explanations. But I do not think that Rashi's explanation is totally implausible/obscene.
Update: In the comments, Chaim B. of Divrei Chaim points out that Tosafot writes about how kiddushei ketana was common practice in the Middle ages. From Kiddushin 41a: