Is there Torah inerrancy?
In other words, does the Torah contain no factual errors: the Noah flood did occur, Garden of Eden did exist, Adam and Eve did live there and were subsequently expelled, etc etc?
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In short the answer is Yes.
Maimonides (also know as the Rambam) codifies 13 principles which are basic to Judaism. These principles are pretty much universally accepted as binding in all Orthodox forms of Judaism. Principle number 8 is, "The belief in the divine origin of the Torah." Principle number 9 is, "The belief in the immutability of the Torah."
This being said, it is important to differentiate between the idea of the Torah being perfect as understood in Judaism and the concept of "Biblical Inerrancy" as understood by many Christian groups.
Judaism believes that the Torah was revealed in two parts. The written text of the Torah was dictated to Moses exactly as we have it today. Together with this "Written Torah", much additional information about each commandment, as well as a complete system of Torah interpretation was was also given to Moses and passed down through the generations.
After the destruction of the Second Holy Temple in 70 CE, a process of collecting and recording these teachings was begun. The teachings of the Oral Torah were eventually codified in the Talmud, Midrash, etc. This process is explained in great detail in the Introduction of the Mishneh Torah by Maimonides.
This means that according to Judaism, one cannot just read the text of the Written Torah and fully understand what God wants from us. One needs to also consult the teachings of the Oral Torah. A classic example is the verse, "and you shall bind them as a sign on your hand and they shall serve as a symbol between your eyes". This verse cannot possibly be understood literally. According to the Oral Torah, it is referring to the practice of Tefillin. There are thousands of detailed rules regarding the preparation and use of Tefillin which are all detailed in the works of the Oral Torah.
This is quite different than the idea of "Biblical Inerrancy" as understood by many Christian groups. Those groups believe that the Bible is to be understood literally at face value. this is completely foreign to Judaism.
This is a matter of opinion, perhaps even amongst Rishonim.
For example, there is Ralbag, who says about the number of stars and Avraham's vision at the brit bein habetarim
לא יחוייב שיהיו אצל הנביא כל הדעות האמיתיות בענין סודות המציאות
Citing a post by Dr. Marc Shapiro:
According to Ralbag, this is an example of a prophet receiving false information in accord with his mistaken conception. Since Abraham falsely believed that there are many stars, his prophecy contained this false conception, while in reality according to Ralbag there are actually a limited number of stars.
As another example (from the same article), from Rav Kook, who refers to Rambam as a basis for the idea:
R. Kook explains that the Torah can describe events in a way not in accord with the astronomical or geological (i.e., historical) truth. This is done in order for the Torah to accomplish its goal, which is not focused on such matters but rather on ידיעת האלהות והמוסר וענפיהם בחיים ובפועל, בחיי הפרט, האומה והעולם
That is, based on the contemporary understanding of the world. The basis in Rambam is in factual errors in Yechezkel, so perhaps it would (according to Rambam) not extend to the Torah.
Another example is (same post) "[t]he fourteenth-century R. Eleazar Ashkenazi ben Nathan ha-Bavli". When discussing the extremely long lifespans, he suggested
that the Torah recorded what the popular belief was, no matter how exaggerated, and Moses was not concerned about these sorts of things. In other words, just like today people say that the Torah is not interested in a scientific presentation of how the world was created, R. Eleazar’s position is that the Torah is not interested in a historically accurate presentation. In his mind, this has nothing to do with the Torah’s goals, and therefore there was no reason for the Torah not to present matters as they were believed at the time, even if these perceptions were inaccurate. The important thing, he says, is that the people would know that from the creation of the world until Israel stood at Sinai was close to three thousand years. This would help solidify belief in creation. The records of lifespans are just a means to illustrate this information. He adds that when it came to events closer to Moses’ time, Moses was careful in preserving a more accurate accounting, while leaving the stories of the distant past cloaked in legend.
For an example of a rishon propounding the idea that the Torah incorporates some necessary false beliefs, see Rabbi Yosef Ibn Caspi. So too Shadal. While I've written about it, so has Dr. Marc Shapiro. See this post on the Seforim blog. For instance:
I must now deal with R. Joseph Ibn Caspi, who is often described as holding a view similar to what we have seen already, but more radical in that he saw it as a general principle of interpretation. I refer to the notion that the Torah incorporates all sorts of untruths because these were what people believed at the time. It is said that this is how Ibn Caspi understands the rabbinic phrase “The Torah speaks in the language of men.”
I would note that various Rishonim are not accepted in the current ultra-Orthodox world. There is a no-true-Scotsman issue:
Person A: "No Scotsman puts sugar on his porridge."
Person B: "But my uncle Angus likes sugar with his porridge."
Person A: "Ah yes, but no true Scotsman puts sugar on his porridge."
Various very religious and scholarly Jews held beliefs that nowadays are not standard frum theology, and the danger in rejecting them as non-Rishonim is that we first define who the Rishonim were based on theology and then assert we are following the Rishonim in matters of theology. (In similar manner, Abarbanel -- who rejected Ibn Caspi -- is himself considered 'not from our beis medrash' and Yes, some contemporaries strongly criticized him for taking philosophical positions that they disagreed with, but so were others criticized who are certainly considered Rishonim (e.g. Ibn Ezra, Rambam).
The dictionary defines inerrancy as "freedom from error", in which case, yes, the Torah has inerrancy. One of the cardinal beliefs of Judaism, according to the Rambam, is that God composed the Pentateuch; the others sorta imply he has inerrancy; combined, then, we get that according to the Rambam the Pentateuch has inerrancy. I have no source at the moment for the other books of Tanach, but I'm pretty sure they, too, have inerrancy.
If your question is, "Can the Torah be wrong?," then the Orthodox will tell you "no" and Conservative and Reform may vary from rabbi to rabbi.
If you're asking "Can the Torah be taken not as face value in regard to historical events?," even in Orthodoxy its a matter of great controversy with many strong;y-worded tshuvos (responsa) and strongly-held opinions.
Typically the way that counterfactual or inaccurate statements in the text (when admitted to be so) are reconciled with inerrancy is by conceding that the text is subject to interpretation and "speaking in the manner of people".
This is a very important question and needs a special treatment to understand its premises. So before we can discuss this question, we need to define some terms:
"Reality": The question is based on a premise that only the physical reality is called "the reality" when the truth is that a lot of Torah interpretation is based on "virtual reality" where the persons and places denote ideas and not real places AKA non-literal reading.
"Truth": The question also implies that only the empirical truth is called the truth, then there are two kinds of truth (as I've mentioned lots of times in question of this kind): De-Jure and De-Facto truth. De-Facto truth is the empirical truth whose truthfulness lies in the correlation between empirical observations, while the De-Jure truth is the truth because "I (or somebody else) said so".
For example, I say my name is Al - is it true? You might say, empirically it isn't, as my Israeli ID (TZ) says Eliyahu. But maybe the TZ lies and it is just a Biblical name I never use and my true name that I call myself is Al?
Same with the Torah - it is inerrant De-Jure, as G-d or Moses or Rambam said so, but it does not have to fit our empirical knowledge.
"Semantics": this stems from #2. We can never be sure we understand the meaning of the words and the author could always claim "you just got it all wrong". For example what "חשמל" is or "סיני" or "עקב" or anything else. Our knowledge is based on the "second-tier" interpretations which are human and defaultively not inerrant. So we can not speak of the inerrancy of the Torah without being able to understand the meaning of its words.
We have multiple of example from the Gemmorah where the Sages didn't know the meaning of the words of the Torah (e.g. Meggila 18a לא הוו ידעי רבנן מאי סירוגין)
To sum it up. It is either YES or NO, depends on who you ask: from G-d's perspective it is YES, and from our empirical perspective, it is NO (or impossible to test).
After a generation of people have debated the Torah Codes,this should not be a question. The letters of the Torah are precise as are the words as are the characters.The details will not be known for now but the entire corpus should be seen by all of us as Divine ,not merely inspired. While there are legitimate concerns of the exactitude of minute detail,this is the best at preserving a text that can be humanly possible over 3323 years since first received.