were those that lived in the old testament times saved by grace or by their works?

None of the patriarchs were perfect, even though they often tried hard to do God's will, so we have to say that it is by grace that they are saved. Moses, having done God's will faithfully spoilt it all by not being obedient in the striking of the rock. Yet he gets approval for his faithfulness in Hebrews as we can see:

Hebrews 11

24 By faith Moses, when he was come to years, refused to be called the son of Pharaoh's daughter;

25 Choosing rather to suffer affliction with the people of God, than to enjoy the pleasures of sin for a season;

26 Esteeming the reproach of Christ greater riches than the treasures in Egypt: for he had respect unto the recompence of the reward.

27 By faith he forsook Egypt, not fearing the wrath of the king: for he endured, as seeing him who is invisible.

28 Through faith he kept the passover, and the sprinkling of blood, lest he that destroyed the firstborn should touch them.

29 By faith they passed through the Red sea as by dry [land]: which the Egyptians assaying to do were drowned.

David arranged the death of Bathsheba's husband so that he could disguise his sin and marry her yet he was called a friend of God because he wanted to live by God's ways.

All sin and fall short of the glory of God but it depends what our hearts are centred on

and whether we try to live by the standards God has set and do what we hope will please Him.

Works & following the Law

I interpret it as in the Old Testament people were saved by their works, because of the sacrifices they had to give to God, all of the rituals and laws they had to follow, and because of the wrath God poured down upon the earth several times when people were not morally sufficient in His eyes. Then Jesus came, taught people that they could never be redeemed by works no matter how well they followed laws like Leviticus, and He took the burden to save humanity with grace. So the Old Testament is about how humanity can never measure up to God, and the New Testament is about accepting that and accepting the grace God gave us with Jesus.

Law.

They were all saved by their works. These saints of OT went to hades temporarily but after Christ rose from the dead the saints of OT went to Paradise.