Is true holiness possible?
Yes. Entire sanctification is a state of perfect love, righteousness, and true holiness which every regenerate believer may obtain.
Catechism for the Global Methodist Church, question 70
6You shall be to me a kingdom of priests and a holy nation. These are the words that you shall speak to the sons of Israel.
44For I am the LORD your God; therefore consecrate yourselves and be holy, for I am holy. Do not defile yourselves with any swarmer that creeps on the land. 45For I am the LORD who brought you up from the land of Egypt to be your God; therefore be holy, for I am holy.
2Speak to all the congregation of the sons of Israel and say to them: You shall be holy, for I the LORD your God am holy.
6For you are a people holy to the LORD your God; the LORD your God has chosen you to be a people for his own possession out of all the peoples that are upon the face of the ground.
2For you are a people holy to the LORD your God. The LORD has chosen you to be a people for a possession to him out of all the peoples that are upon the face of the earth.
43You have heard that it was said, You shall love your neighbor and you shall hate your enemy. 44But I say to you, Love your enemies and pray for those who persecute you. 45That you may become sons of your Father who is in the heavens, for he causes his sun to rise upon evil ones and upon good ones, and he causes rain to fall upon just ones and upon unjust ones. 46For if you love those who love you, what reward do you have? Do not even the tax collectors do the same? 47And if you greet only your brothers, what more are you doing? Do not even the Gentiles do the same? 48Therefore you shall be perfect, as your heavenly Father is perfect.
2Beloved, now we are children of God, and what we shall be has not yet been revealed. We know that when he is revealed we shall be like him, because we shall see him as he is. 3And everyone who has this hope in him purifies himself just as that one is pure.
3For this is the love of God, that we keep his commandments; and his commandments are not burdensome.

With this question the catechism moves from sanctification in general to the distinctive Methodist teaching of entire sanctification, which Wesley also called Christian perfection. The answer is quoted from the second paragraph of Article XI of the Confession of Faith: "Entire sanctification is a state of perfect love, righteousness and true holiness which every regenerate believer may obtain." ΒΆ102.6 of the Book of Doctrines and Discipline names the same doctrine as the "fullness of salvation" and says the church holds it "with John Wesley."
The Yes rests on the character of God's commands. Scripture never treats holiness as an ideal beyond reach. The Lord tells Israel, "You shall be to me a kingdom of priests and a holy nation" (Exodus 19:6), and in Leviticus the command comes with its reason: "You shall be holy, for I the LORD your God am holy" (Leviticus 19:2; compare Leviticus 11:44-45). Deuteronomy states the same as an accomplished fact, "you are a people holy to the LORD your God" (Deuteronomy 7:6; 14:2). Jesus gathers all of it into one sentence at the close of the Sermon on the Mount: "Therefore you shall be perfect, as your heavenly Father is perfect" (Matthew 5:48). Wesley's argument was simple: God does not command what his grace will not give.
The answer's phrase is perfect love, and Matthew 5:43-48 shows what it means. The perfection Jesus describes is love for enemies, prayer for persecutors, and a goodness that, like sunshine and rain, falls on the just and the unjust alike. Christian perfection is therefore a perfection of love and intention, and the next question will say so directly. It is true holiness, a real cleansing of the heart, and it is available to every regenerate believer, since it is a work of grace received by faith and is the ordinary end of the new birth. John writes that "everyone who has this hope in him purifies himself just as that one is pure" (1 John 3:3) and that God's "commandments are not burdensome" (1 John 5:3).
For a Methodist this teaching sets the expectation for the whole Christian life. A believer is meant to hope for a heart in which love governs everything, and to ask God for it plainly. Wesley held that this gift is God's to give and ours to seek, and the following questions describe how it is received, how it is kept, and what it does and does not promise.