Was eternal life promised by the Mosaic Law?

No. Moses didn't write anything about eternal life or a future resurrection. Both of these were new ideas that were invented much later by the Iranian prophet Zoroaster. More

To think that the Mosaic law did not promise eternal life implies that it did not provide eternal life, or salvation or justification. This insinuates that the old saints are not saved but went to hell.

The OT may not contain the NT term "eternal life" but whenever it promised "life" it meant exactly the same.

Daniel (12:1-2) 1 “At that time shall arise Michael, the great prince who has charge of your people. And there shall be a time of trouble, such as never has been since there was a nation till that time. But at that time your people shall be delivered, everyone whose name shall be found written in the book. 2 And many of those who sleep in the dust of the earth shall awake, some to everlasting life, and some to shame and everlasting contempt.

Deut (4: 1-2) “And now, O Israel, listen to the statutes and the rules that I am teaching you, and do them, that you may live, and go in and take possession of the land that the LORD, the God of your fathers, is giving you. 2 You shall not add to the word that I command you, nor take from it, that you may keep the commandments of the LORD your God that I command you. ESV

Deut (10: 12-13) 12 “And now, Israel, what does the LORD your God require of you, but to fear the LORD your God, to walk in all his ways, to love him, to serve the LORD your God with all your heart and with all your soul, 13 and to keep the commandments and statutes of the LORD, which I am commanding you today for your good?

Lev 18:5 You shall therefore keep my statutes and my rules; if a person does them, he shall live by them: I am the LORD.

Deut 30:16 If you obey the commandments of the LORD your God that I command you today, by loving the LORD your God, by walking in his ways, and by keeping his commandments and his statutes and his rules, then you shall live and multiply, and the LORD your God will bless you in the land that you are entering to take possession of it.

All Jews believe that the law gives eternal life, even though some rabbinic Jews might not use the term.

Rashi commentary on Lev 18:5 -- You shall observe My statutes and My ordinances: [This comes to include other details of [laws pertaining to] this passage, that Scripture did not mention explicitly. — [Torath Kohanim 18:142] Another explanation: [This clause, ''You shall observe My statutes and My judgments, which a man shall do,'' was added,] in order to apply ''observance'' (שְׁמִירָה) and fulfillment (עֲשִׂיָּה) to statutes, and to apply ''observance'' (שְׁמִירָה) and ''fulfillment'' (עֲשִׂיָּה) to ordinances. For [up till now,] Scripture had mentioned only ''fulfillment'' regarding ordinances and ''observance'' regarding statutes (see previous verse).

• [Torath Kohanim 18:134]and live by them: in the World-To-Come. For if you say [that the verse refers to living] in this world, does he not eventually die? [Torath Kohanim 18:134]I am the Lord: faithful to pay a reward [an attribute represented by the Name ה ]. — [Torath Kohanim 18:134]

Palestine Targum renders Lev 18:5 as

And you shall keep My statutes, and the order of My judgments, which if a man do he shall live in them, in the life of eternity, and his portion shall be with the just: I am the Lord.

Following are some excerpts from an article on Book of Life

"[T]hree books are opened in heaven on Rosh Ha-Shanah, one for the thoroughly wicked, one for the thoroughly righteous, and one for the intermediate. The thoroughly righteous are forthwith inscribed in the Book of Life, the thoroughly wicked in the Book of Death, while the fate of the intermediate is suspended until the Day of Atonement" (Rosh HaShanah 16b).

(Encyclopaedia Judaica, 1997 CD-ROM edition)

ספר הײם (Ps. 69:29, "book of the living"; so LXX., Vulg., R.V. [margin]). The living are the righteous (second half of the verse), who alone are admitted to citizenship in the theocracy. The wicked are denied membership therein: they are blotted out of God's book (Ex. 32:32 et seq.). The figure is derived from the citizens' registers (Ezek. 13:9; Jer. 22:30; and Ex. 32:30-34, accordingly assigned by Holzinger to a late. stratum; see his commentary). The life which the righteous participate in is to be understood in a temporal sense. In Dan. xii. 1, however, those who are found written in the book and who shall escape the troubles preparatory to the coming of the Messianic kingdom are they who, together with the risen martyrs, are destined to share in the everlasting life referred to in verse 2. The eternal life is certainly meant in Enoch xlvii. 3, civ. 1, cviii. 3, and frequently in the New Testament (especially in Revelation). The Targum (Isa. 4:3; Ezek. 13:9) speaks of the "Book of Eternal Life." Temporal life is apparently prayed for in the liturgical formula: "Inscribe us in the Book of Life" (see Atonement, Day of). The Mishnah tells us that the deeds of every human being are recorded in a book (Abot, ii. 1; see iii. 16). The "Sefer Chasidim" (xxxiii.) pointedly adds that God is in no need of a book of records; "the Torah speaks the language of man"; i.e., figuratively.

Paul also quoted Lev 18:5 in Romans 10:5 and Galatians 3:12

David Stern's Jewish New Testament commentary on Romans 10:5

5 The quotation is from Leviticus 18:5, “You shall therefore keep my statutes and my judgments, which, if a person does them, he will live by them.” Should there be any doubt whether the righteousness that results from obeying God’s statutes and judgments leads to eternal life, the verse says, “he will live by them,” on which Rashi (quoting the Sifra, a fourth-century collection of midrashim related to Leviticus), comments: “It refers to the world to come; for if you say it refers to this world, doesn’t everyone die sooner or later?” Thus I translate the phrase, he will attain life through them—eternal life. The word for “live” or “attain life” is the same as that used at 8:12–13 to describe what will happen to the believer who “by the Spirit” keeps “putting to death the practices of the body.” Conclusion: Sha’ul affirms that the Torah and the Ruach HaKodesh offer one and the same eternal life. This is consistent with and suggested by the fact that the Holy Spirit came to the first believers on Shavuot (Pentecost), the same day the Torah was given to Moses (see Ac 2:4&N). The two most important of the “statutes and judgments” referred to in Leviticus 18:5 are stated by Yeshua at Mark 12:28–31: (1) loving God (the Shema, Deuteronomy 6:4–5) and (2) loving one’s neighbor as oneself (Leviticus 19:18). Both are predicated on trusting in God: you can’t love God if you don’t believe in him as who he says he is, and since both you and your neighbor are made in God’s image you can’t love your neighbor as yourself in the sense that the Torah demands without believing in the God who made both of you. Therefore, Leviticus 18:5, quoted here by Sha’ul, backs up his point that obeying the Torah requires trust, not legalistic works.

Notice especially this passage where Jesus implied that you must fulfill the law to inherit eternal life. The New Covenant was not instituted at that time.

(Luke 10:25-28) 25 And behold, a lawyer stood up to put him to the test, saying, “Teacher, what shall I do to inherit eternal life?” 26 He said to him, “What is written in the Law? How do you read it?” 27 And he answered, “You shall love the Lord your God with all your heart and with all your soul and with all your strength and with all your mind, and your neighbor as yourself.” 28 And he said to him, “You have answered correctly; do this, and you will live.”

(Matthew 19:16-19) 16 And behold, a man came up to him, saying, “Teacher, what good deed must I do to have eternal life?” 17 And he said to him, “Why do you ask me about what is good? There is only one who is good. If you would enter life, keep the commandments.” 18 He said to him, “Which ones?” And Jesus said, “You shall not murder, You shall not commit adultery, You shall not steal, You shall not bear false witness, 19 Honor your father and mother, and, You shall love your neighbor as yourself.”

James Dunn writes in note 23 of chapter 21 of his book The New Perspective on Paul, (If someone knows the exact book names of some of the references cited here, then add them here):

The standard understanding of Lev. 18.5, as inculcating a life lived by doing the commandments, is indicated by Ezek. 20.5–26 (the earliest commentary on Lev. 18.5);as confirmed by Deut. 30.15–20;Prov. 3.1–2;6.23;Neh. 9.29;Bar. 4.1;1QS 4.6–8;CD 3.15–17;Pss. Sol. 14.2–3; Ep. Arist. 127;Philo, Cong. 86–87; LAB 23.10;4 Ezra 7.21;m. Abot 2.7;’the law of life’ (Sir. 17.11;45.5;4 Ezra 14.30);‘the commandments of life’ (Bar. 3.9). The thought seems to have emerged (in Qumran?) that the length of life thus promised could be expressed in terms of (acquiring) ‘eternal life’ (1QS 4.7 – ‘plentiful peace in a long life … eternal enjoyment with endless life’;CD 3.20 – he ‘will acquire eternal life’; 7.6 – ‘they shall live a thousand generations’);see again m. Abot 2.7. The thought of resurrection to eternal life (Dan. 12.2;2 Macc. 7.9;cf. 4 Macc. 15.3) does not appear to be so directly connected with Lev. 18.5. S.J.

It is correct that the written Hebrew Scriptures, aka the "Old Testament", says nothing specific about the afterlife. For that reason, the Sadduccees held that there was no life after death and no punishment for sins after death. Mark 12:18.

The Pharisees, in contrast, believed in an afterlife and believed that the concept had been retained in the "Oral Law." According to the rabbis, what we know of G-d's Law (i.e. the Torah) that was given at Mount Sinai is contained in the "Written Torah" (the Five Books of Moses -- Genesis through Deuteronomy) and in the "Oral Torah" which were detailed explanations about the commandments and theology sometimes alluded to in the Written Torah but not stated explicitly. The Written Torah itself hints at this at several places, most obviously at Deut. 12:21, where G-d says "Then you may slaughter of your herd and flock...as I have commanded you." Nowhere in the written Torah is there any commandment regarding how to slaughter. From this verse we know that the method was taught by G-d to Moses and he, in turn, orally taught those details to Joshua and the Elders, who in turn passed it on to their successors, and so on. The Sadducces rejected the existence of the Oral Torah. Josephus, Antiquities 13.10.6. Jesus, it would appear, accepted the Pharisee's understanding of the Oral Torah for he said that the Pharisees "sit in the seat of Moses; therefore all they tell you, do and observe." Matt. 22:2-3.

The Pharisees wrote down the Oral Law in the years after the Second Temple was destroyed because the Romans had done much to try to murder the rabbis and thereby destroy knowledge of the Oral Law. Iggeres of Rav Shireria Gaaon (ca 982 Babylon). In their writings, they frequently make mention of the World to Come in terms of both life after death and in terms of the resurrection during the Messianic Era. With reference to the former, the Mishna Avos 4:16-17 quotes Rabbi Yaakov saying: "This world is comparable to the antechamber before the World to Come. Prepare yourself in the antechamber, so that you may enter the banquet hall," and"A single moment of repentance and good deeds in this world is greater than all of the World to Come. And a single moment of bliss in the World to Come is greater than all of the present world."

The Pharisees also compared the holiness and tranquility of Heaven with that of a properly-observed Sabbath: "Whoever toiled on the eve of the Sabbath (i.e. Friday) will eat on the Sabbath itself, but whoever did not toil on the eve of the Sabbath, from where will he eat on the Sabbath?" Talmud Bavli, Avodah Zarah 3a (i.e. our efforts in this world are like the efforts we make to prepare each week for the Sabbath). But, as this quotation suggests, the Rabbis would agree that observance of the Torah laws is a prerequisite for getting into Heaven and our efforts in this World is to increase and improve our observance of G-d's commandments so that we will be prepared for the world to come. See, e.g. Luzatto, Moshe Chaim, The Complete Mesillat Yesharim (Ofeq Institute 2007) p. 305 (originally published in 1740).