This seems to be an internet myth. There are plenty of websites that claim the Bible says “Slavery is an everlasting institution”, and they normally cite Leviticus 25:44-46. But here is what those verses really say:
44 As for your male and female slaves whom you may have: you may buy male and female slaves from among the nations that are around you. 45 You may also buy from among the strangers who sojourn with you and their clans that are with you, who have been born in your land, and they may be your property. 46 You may bequeath them to your sons after you to inherit as a possession forever. You may make slaves of them, but over your brothers the people of Israel you shall not rule, one over another ruthlessly.
Another related passage is Deuteronomy 15:12-18:
12 “If your brother, a Hebrew man or a Hebrew woman, is sold to you, he shall serve you six years, and in the seventh year you shall let him go free from you. 13 And when you let him go free from you, you shall not let him go empty-handed. 14 You shall furnish him liberally out of your flock, out of your threshing floor, and out of your winepress. As the Lord your God has blessed you, you shall give to him. 15 You shall remember that you were a slave in the land of Egypt, and the Lord your God redeemed you; therefore I command you this today. 16 But if he says to you, ‘I will not go out from you,’ because he loves you and your household, since he is well-off with you, 17 then you shall take an awl, and put it through his ear into the door, and he shall be your slave forever. And to your female slave you shall do the same. 18 It shall not seem hard to you when you let him go free from you, for at half the cost of a hired servant he has served you six years. So the Lord your God will bless you in all that you do.
So the Bible provided laws for ancient Israel in order to control slavery. But nowhere does it encourage slavery, and nowhere does it say it is an “everlasting institution”.