Take your Bible (I assume you have one) and open it to Deuteronomy, Chapter 5, verse 29 (I am using a Hebrew Tanach. The KJV may have the chapters numbered differently)
5-29. You shall ve careful to act as the Lord, your God, commanded you, you shall not stray to the right or left.
30. On the entire way that the Lord, your God, commanded you shall you go, so that you shall live and it will be good for you, and you shall prolong your days in the Land that you shall inherit.
6-1. This is the commandment, and the statutes, and the judgments that the Lord, your God, commanded to teach you, to perform in the Land into which you are crossing, to inherit it,
2. So that you will fear the Lord, your God, to observe all his statutes and commandments that I command you--you, your child, and your grandchild--all the days of your life, so that your days will be lengthened.
The repetition of "all the days of your life" and "length of days" in verse 6-2 refers to eternal life.
How is someone deemed righteous? I have also heard that the righteous does not necessarily include only Jews. Does Judaism teach that "salvation" is possible apart from keeping the Jewish law?
All Jews are expected to observe the Law. Gentiles are not obligated in the ceremonial or the ritual law but can achieve salvation by acknowledging that G-D is the creator and rulers the world and by upholding justice and morality.
Here you go.
See also the article linked therein.