The gemorah in Bava Basra 91b indicates that Rus (Ruth) was still alive at the time of Shlomo Hamelech. Recognizing that not all agadeta is to be taken literally (in this instance, for example, see Radak on Melachim 1;2, 19; but see Maharsha on BB 91b and cf Rashi on Melachim 1;2, 19) how do we know if this maamar is or is not, and if not, what hidden lesson or point is the gemorah trying to impart?
(a) With regard to the longevity of Ruth, while there is no reason in principle that she should not have seen her great-great-grandson, it is highly unlikely according to the normal rules of nature, as Radak (Melachim I 2:19) points out, especially as we know that David was the youngest son of Yishai (Shmuel I 16: 10-11) and that Shlomo was David's youngest son, born when he was already a mature king (see Shmuel II 5:4 and 5 and 14).
Altogether, the chronology of Boaz and Ruth is highly complicated with numerous Midrashim in disagreement with each other. For example, while we usually assume that the story of Ruth occurred during the time of the later Shoftim at the time of Ivtzan, just prior to Shimshon, Ruth Rabah 1:1 places the whole episode centuries earlier, either at the time of Ehud/Shamgar, or the time of Devorah/Barak. Ruth Rabah 2:9 and Yalkut on Shoftim 3:20 say that Ruth was the daughter (the Gemara in Nazir 23b says granddaughter ) of Eglon, King of Moav (Shoftim 3:12), who was killed by Ehud centuries earlier than Ivtzan (= Boaz, according to Bava Basra 91a). See, however, Tosfos in Nazir 23b (DH Bas), who says that the Gemara is not to be taken literally as granddaughter, but rather descendant.
Ruth Rabah 4:4 tells us that Ruth was only 40 when she married Boaz. Our Gemara (Bava Basra 91b, as well as Ruth Rabah 2:2 and Yalkut Shimoni (Melachim #170 and Divrei Hayamim #1075) holds that the throne provided by Shlomo for the "Ima Shel Malchus," King-mother, was for Ruth. If it agrees with the Ruth Rabah 1:1 mentioned above (and Seder Olam Rabah #12) that she lived in the time of Shamgar, she must have been almost 400 years old at the time of Shlomo, and even if Bava Basra 91b goes only like the usual opinion that Boaz is to be identified with Ivtzan, then we still have a period of 130 years from his death when Ruth was 40 until the reign of Shlomo, making her 170 years old.
Quite apart from this, there are major problems in fitting the 10 generations from Peretz to David without assuming terrific longevity somewhere. Nachshon went into the Red Sea but did not enter Eretz Yisrael (according to Seder Olam he died in the second year in the wilderness). If so, either Salmon (his son) or Boaz (his grandson) must have been terrifically old. See Rashi Divrei Hayamim I 2:11 that Boaz would have had to be 300 years old. Similar problems arise with the identification of Orpah as the mother of Goliath.
In fact, the question of chronology in the case of Ruth can be put even more simply. We know that Chetzron came down to Egypt (in Parshas Vayigash). From then until the Beis Hamikdash was 690 years (i.e. 210 in Egypt (= "ReDU") + 480 years from the Exodus until the building of the Beis Hamikdash, Melachim I 6:1) during which time nine generations were born; see Ibn Ezra (end of Ruth) that this would have Ruth giving birth at the age of 91. The Ramban, in a famous comment (Bereishis 46:15), points out this chronological problem and writes that it was because of this that Chazal attributed to Ovad an exceptionally long life. This, he writes, was a "Nes Nistar," a hidden miracle, which Hash-m performed for the "father of the kingship of Israel, son of the righteous one who chose to join the Jewish nation (= Ruth)." He adds that Chazal also attributed to Ruth extreme longevity based on this chronological problem.
Thus, while Agados need not be taken literally, here the verses themselves cry out for explanation and the Ramban is quite clear that a miracle was involved. Even the logical and pragmatic Ibn Ezra resorts to what we might consider unlikely chronology. In summation, with regard to the particular Gemara in question, from the foregoing it is evident that there is strong basis to take it literally.
Extreme longevity is not unheard of among the righteous. Even the Rambam (who is known to be most rationalistic) holds in his introduction to Mishneh Torah that Eli learnt Torah from Pinchas, which makes the latter extremely old indeed, and that Achiyah ha'Shiloni came out of Egypt and was still alive at the time of Eliyahu -- about 600 years later!
(b) On the other hand, one could explain that the Gemara's intention here is that Shlomo provided a special chair in honor of Ruth (i.e. to remind everyone of his origins, as we will explain). The verse from Divrei Hayamim I 4:23, "Yashvu Sham," which the Gemara says refers to Ruth, could be interpreted in the same way. Though Maharsha suggests that the plural ("Yashvu") is to include both Ruth and Batsheva, it could still be interpreted as including Ruth only in this allegorical way, namely, that her presence and influence was felt as if she was seated with them -- the chair in her honor made sure of this. The words at the end of the second verse cited by the Gemara, "... and she sat to his right," refer not to Ruth but to Batsheva, Shlomo's own mother (see Maharzu on Ruth Rabah 2:2).
Why would Shlomo put a reminder of his illustrious great-great-grandmother in the throne-room? We can find the answer to this in the comments of the Ben Yehoyada here. The Ben Yehoyada asks a number of questions: (1) When did the verse describe Ruth as "Em ha'Melech?" She was neither his mother nor his grandmother! (2) Why did the Gemara describe Ruth as " Imah Shel Malchus" and not more simply as " Em ha'Malchus"?
He answers that Chazal say Shaul's rule did not last since he had no "impurity" in his genealogy, thus leading to arrogance (see Yoma 22b). David, however, achieved permanence because of the fact that he stemmed from Ruth, enabling him to retain his humility. Thus, Ruth was the mother and source of his ability to retain the throne! Perhaps Shlomo wanted to retain a permanent reminder of his origin and he prepared a vacant chair with Ruth's name etched on it for this purpose.
Although this method of explanation of Agadah is perfectly valid, on this occasion there is no real need for it if we follow the Ramban, as explained above.