A popular myth is that Isaiah 53 was never considered messianic by rabbis and Jewish sages. Judaism teaches that Isaiah 53 refers to the nation of Israel.
Isaiah 53 (more precisely, 52:13 to 53:12) has been interpreted in messianic terms by a wide variety of Jewish commentators over a long period of time. Other interpretations have certainly been offered, including the view first popularized by Rashi in medieval times that the prophet speaks of the nation of Israel. Neverthless the messianic interpretation has a long history in Jewish Bible exegesis, as shown by the quotations below.
Behold, My Servant the Messiah shall prosper.
-- Targum ("Targum Jonathan") to Isaiah 52:13, various editions (such as Samson H. Levey, The Messiah: An Aramaic Interpretation; the Messianic Exegesis of the Targum." Cincinnati: Hebrew Union College, 1974, p. 63).
In the early cycle of synagogue readings
We know that messianic homilies based on Joseph's career (his saving role preceded by suffering), and using Isaiah 53 as the prophetic portion, were preached in certain old synagogues which used the triennial cycle...
-- Rav Asher Soloff, "The Fifty Third Chapter of Isaiah According to the Jewish Commentators, to the Sixteenth Century" (Ph.D. Thesis, Drew University,1967), p. 146.
The addition of 53.4-5 [to the cycle of synagogue readings] was evidently of a Messianic purport by reason of the theory of a suffering Messiah. The earlier part of [the Haftarah] (52.7ff.) dealt with the redemption of Israel, and in this connection the tribulations of the Messiah were briefly alluded to by the recital of the above 2 verses.
-- Jacob Mann, The Bible as Read and Preached in the Old Synagogue (NY: Ktav, 1971, © 1940), p. 298.
The Rabbis said: His name is "the leper scholar," as it is written, Surely he hath borne our griefs, and carried our sorrows: yet we did esteem him a leper, smitten of God, and afflicted. [Isaiah 53:4].
-- Soncino Talmud edition.
The fifth interpretation [of Ruth 2:14] makes it refer to the Messiah. Come hither: approach to royal state. And eat of the BREAD refers to the bread of royalty; AND DIP THY MORSEL IN THE VINEGAR refers to his sufferings, as it is said, But he was wounded because of our transgressions. (Isa. LIII, 5).
-- Soncino Midrash Rabbah (vol. 8, p. 64).
The Karaite Yefeth ben Ali (10th c.)
As to myself, I am inclined, with Benjamin of Nehawend, to regard it as alluding to the Messiah, and as opening with a description of his condition in exile, from the time of his birth to his accession to the throne: for the prophet begins by speaking of his being seated in a position of great honour, and then goes back to relate all that will happen to him during the captivity. He thus gives us to understand two things: In the first instance, that the Messiah will only reach his highest degree of honour after long and severe trials; and secondly, that these trials will be sent upon him as a kind of sign, so that, if he finds himself under the yoke of misfortunes whilst remaining pure in his actions, he may know that he is the desired one....
-- S. R. Driver and A. Neubauer, editors, The Fifty-third Chapter of Isaiah According to the Jewish Interpreters (2 volumes; New York: Ktav, 1969), pp. 19-20. The English translations used here are taken from volume 2. The original texts are in volume 1. Cf. Soloff, pp. 107-09.
Another statement from Yefeth ben Ali:
By the words "surely he hath carried our sicknesses," they mean that the pains and sickness which he fell into were merited by them, but that he bore them instead. . . . And here I think it necessary to pause for a few moments, in order to explain why God caused these sicknesses to attach themselves to the Messiah for the sake of Israel. . . . The nation deserved from God greater punishment than that which actually came upon them, but not being strong enough to bear it. . . God appoints his servant to carry their sins, and by doing so lighten their punishment in order that Israel might not be completely exterminated.
-- Driver and Neubauer, pp. 23 ff.; Soloff pp. 108-109.
Another statement from Yefeth ben Ali:
"And the Lord laid on him the iniquity of us all." The prophet does not by avon mean iniquity, but punishment for iniquity, as in the passage, "Be sure your sin will find you out" (Num. xxxii. 23).
-- Driver and Neubauer, p. 26; Soloff p. 109.
Mysteries of R. Shim'on ben Yohai (midrash, date uncertain)
And Armilaus will join battle with Messiah, the son of Ephraim, in the East gate . . .; and Messiah, the son of Ephraim, will die there, and Israel will mourn for him. And afterwards the Holy One will reveal to them Messiah, the son of David, whom Israel will desire to stone, saying, Thou speakest falsely; already is the Messiah slain, and there is non other Messiah to stand up (after him): and so they will despise him, as it is written, "Despised and forlorn of men;" but he will turn and hide himself from them, according to the words, "Like one hiding his face from us."
-- Driver and Neubauer, p. 32, citing the edition of Jellinek, Beth ha-Midrash (1855), part iii. p. 80.
Lekach Tov (11th c. midrash)
"And let his [Israel's] kingdom be exalted," in the days of the Messiah, of whom it is said, "Behold my servant shall prosper; he will be high and exalted, and lofty exceedingly."
-- Driver and Neubauer, p. 36.
Maimonides, Letter to Yemen (12th c.)
What is to be the manner of Messiah's advent, and where will be the place of his appearance? . . . And Isaiah speaks similarly of the time when he will appear, without his father or mother of family being known, He came up as a sucker before him, and as a root out of the dry earth, etc. But the unique phenomenon attending his manifestation is, that all the kings of the earth will be thrown into terror at the fame of him -- their kingdoms will be in consternation, and they themselves will be devising whether to oppose him with arms, or to adopt some different course, confessing, in fact, their inability to contend with him or ignore his presence, and so confounded at the wonders which they will see him work, that they will lay their hands upon their mouth; in the words of Isaiah, when describing the manner in which the kings will hearken to him, At him kings will shut their mouth; for that which had not been told them have they seen, and that which they had not heard they have perceived.
-- Driver and Neubauer vol 1: p. 322. Edition is Abraham S. Halkin, ed., Igeret Teman (NY: American Academy for Jewish Research, 1952). See Soloff pp. 127-128.
Zohar II, 212a (medieval)
There is in the Garden of Eden a palace named the Palace of the Sons of Sickness. This palace the Messiah enters, and He summons every pain and every chastisement of Israel. All of these come and rest upon Him. And had He not thus lightened them upon Himself, there had been no man able to bear Israel's chastisements for the transgressions of the law; as it is written, "Surely our sicknesses he has carried."
-- Cited in Driver and Neubauer, pp. 14-15 from section "va-yiqqahel". Translation from Frydland, Rachmiel, What the Rabbis Know About the Messiah (Cincinnati: Messianic Literature Outreach, 1991), p. 56, n. 27. Note that this section is not found in the Soncino edition which says that it was an interpolation.
Nachmanides (R. Moshe ben Nachman) (13th c.)
The right view respecting this Parashah is to suppose that by the phrase "my servant" the whole of Israel is meant. . . .As a different opinion, however, is adopted by the Midrash, which refers it to the Messiah, it is necessary for us to explain it in conformity with the view there maintained. The prophet says, The Messiah, the son of David of whom the text speaks, will never be conquered or perish by the hands of his enemies. And, in fact the text teaches this clearly. . . .
And by his stripes we were healed -- because the stripes by which he is vexed and distressed will heal us; God will pardon us for his righteousness, and we shall be healed both from our own transgressions and from the iniquities of our fathers.
-- Driver and Neubauer, pp. 78 ff.
Who art thou, O great mountain (Zech. iv. 7.) This refers to the King Messiah. And why does he call him "the great mountain?" Because he is greater than the patriarchs, as it is said, "My servant shall be high, and lifted up, and lofty exceedingly" -- he will be higher than Abraham, . . . lifted up above Moses, . . . loftier than the ministering angels.
-- Driver and Neubauer, p. 9.
The same passage is found in Midrash Tanhuma to Genesis (perhaps 9th c.), ed. John T. Townsend (Hoboken, NJ: Ktav, 1989), p. 166.
Yalkut ii. 620 (13th c.), in regard to Psalm 2:6
I.e., I have drawn him out of the chastisements. . . .The chastisements are divided into three parts: one for David and the fathers, one for our own generation, and one for the King Messiah; and this is that which is written, "He was wounded for our transgressions," etc.
-- Driver and Neubauer, p. 10.
This Parashah the commentators agree in explaining of the Captivity of Israel, although the singular number is used in it throughout. . . .As there is no cause constraining us to do so, why should we here interpret the word collectively, and thereby distort the passage from its natural sense?. . . As then it seemed to me that the doors of the literal interpretation of the Parashah were shut in their face, and that "they wearied themselves to find the entrance," having forsaken the knowledge of our Teachers, and inclined after the "stubbornness of their own hearts," and of their own opinion, I am pleased to interpret it, in accordance with the teaching of our Rabbis, of the King Messiah, and will be careful, so far as I am able, to adhere to the literal sense.
-- Driver and Neubauer, pp. 99-100.
Another comment from R. Mosheh Kohen ibn Crispin
If his soul makes itself into a trespass-offering, implying that his soul will treat itself as guilty, and so receive punishment for our trespasses and transgressions.
-- Driver and Neubauer, p. 112.
R. Sh'lomoh Astruc (14th c.)
My servant shall prosper, or be truly intelligent, because by intelligence man is really man -- it is intelligence which makes a man what he is. And the prophet calls the King Messiah my servant, speaking as one who sent him. Or he may call the whole people my servant, as he says above my people (lii. 6): when he speaks of the people, the King Messiah is included in it; and when he speaks of the King Messiah, the people is comprehended with him. What he says then is, that my servant the King Messiah will prosper.
-- Driver and Neubauer, p. 129.
R. Elijah de Vidas (16th c.)
Since the Messiah bears our iniquities which produce the effect of His being bruised, it follows that whoso will not admit that the Messiah thus suffers for our iniquities, must endure and suffer for them himself.
-- Driver and Neubauer, p. 331.
I may remark, then, that our Rabbis with one voice accept and affirm the opinion that the prophet is speaking of the King Messiah, and we ourselves also adhere to the same view.
-- Driver and Neubauer, p. 258.
The fact is, that it refers to the King Messiah, who will come in the latter days, when it will be the Lord's good pleasure to redeem Israel from among the different nations of the earth.....Whatever he underwent was in consequence of their own transgression, the Lord having chosen him to be a trespass-offering, like the scape-goat which bore all the iniquities of the house of Israel.
-- Driver and Neubauer, p. 400-401.
Our righteous anointed is departed from us: horror hath seized us, and we have non to justify us. He hath borne the yoke of our iniquities, and our transgression, and is wounded because of our transgression. He beareth our sins on his shoulder, that he may find pardon for our iniquities. We shall be healed by his wound, at the time that the Eternal will create him (the Messiah) as a new creature. O bring him up from the circle of the earth. Raise him up from Seir, to assemble us the second time on Mount Lebanon, by the hand of Yinnon.
-- A. Th. Philips, Machzor Leyom Kippur / Prayer Book for the Day of Atonement with English Translation; Revised and Enlarged Edition (New York: Hebrew Publishing Company, 1931), p. 239. The passage can also be found in, e.g., the 1937 edition. Also, Driver and Neubauer, p. 399.