Both the Old and New Testaments, or Covenants, required a blood sacrifice. The author of Hebrews speaks about this:
Hebrews 9:11 But when Christ appeared as a high priest of the good things that have come, then through the greater and more perfect tent (not made with hands, that is, not of this creation) 12 he entered once for all into the holy places, not by means of the blood of goats and calves but by means of his own blood, thus securing an eternal redemption. 13 For if the blood of goats and bulls, and the sprinkling of defiled persons with the ashes of a heifer, sanctify for the purification of the flesh, 14 how much more will the blood of Christ, who through the eternal Spirit offered himself without blemish to God, purify our conscience from dead works to serve the living God.
15 Therefore he is the mediator of a new covenant, so that those who are called may receive the promised eternal inheritance, since a death has occurred that redeems them from the transgressions committed under the first covenant. 16 For where a will is involved, the death of the one who made it must be established. 17 For a will takes effect only at death, since it is not in force as long as the one who made it is alive. 18 Therefore not even the first covenant was inaugurated without blood. 19 For when every commandment of the law had been declared by Moses to all the people, he took the blood of calves and goats, with water and scarlet wool and hyssop, and sprinkled both the book itself and all the people, 20 saying, “This is the blood of the covenant that God commanded for you.” 21 And in the same way he sprinkled with the blood both the tent and all the vessels used in worship. 22 Indeed, under the law almost everything is purified with blood, and without the shedding of blood there is no forgiveness of sins.
23 Thus it was necessary for the copies of the heavenly things to be purified with these rites, but the heavenly things themselves with better sacrifices than these. (ESV)
It is possible to obtain quite a bit of blood from an animal, or a human, without loss of life. The sacrifices mentioned above were more than mere blood sacrifices. They required that the sacrificial animal, or Christ, be killed.
Why a blood sacrifice? I'm not sure, but will speculate.
Blood is the life, according to the Old Testament. (Genesis 9:4, Leviticus 17:11, 14, Deuteronomy 12:23.) What does that mean? Much, maybe all, of the meaning is religious or symbolic. However, there are biological ways in which blood is life, too:
Blood carries Oxygen and food molecules to the body, thus giving life in an even more fundamental way than breathing. Blood also carries metabolic waste products to the kidneys, which remove them. This is also a requirement for life. Without this function, we would die in our poisonous metabolic byproducts.
Blood carries hormones, signals from one part of the body to another. It also contains antibodies, molecules which fight off infections. It has white blood cells, which also defend us. It carries molecules which can clot, plugging holes in our skin or blood vessels.
Perhaps for these reasons, blood has been used as a sacrifice for sin. In the New Testament, Christ's blood also paid the price necessary to defeat death, itself.
Here's a fictional treatment of this:
And there, on the golden gravel of the bed of the stream, lay King Caspian, dead, with the water flowing over him like liquid glass. His long white beard swayed in it like water-weed. And all three stood and wept. Even the Lion wept: great Lion-tears, each tear more precious than the Earth would be if it was a single solid diamond. . . .
"Son of Adam," said Aslan, "go into that thicket and pluck the thorn that you will find there, and bring it to me."
Eustace obeyed. The thorn was a foot long and sharp as a rapier.
"Drive it into my paw, son of Adam," said Aslan. . . .
"Must I?" said Eustace.
"Yes," said Aslan.
Then Eustace set his teeth and drove the the thorn into the Lion's paw. And there came out a great drop of blood, redder than all the redness that you have ever seen or imagined. And it splashed into the stream over the dead body of the King. . . . And the dead King began to be changed. His white beard turned to grey, and from grey to yellow, and got shorter and vanished altogether; and his sunken cheeks grew round and fresh, and the wrinkles were smoothed, and his eyes opened, and his eyes and lips both laughed, and suddenly he leaped up and stood before them--a very young man, or a boy. . . . And he rushed to Aslan and flung his arms as far as they would go round the huge neck; and he gave Aslan the strong kisses of a King, and Aslan gave him the wild kisses of a Lion. -C. S. Lewis, The Silver Chair, New York: Macmillan, 1953, pp. 203-4.