If God is a loving God, full of goodness, righteousness and holiness, why does hell exist?
The short answer is that God created free will, which means we have the right to accept or reject God. Hell happens because it is the natural and logical consequence of rejecting God. Hell is basically the absence of God.
These attributes and gifts from God are absent in hell:
1 John 1:5 King James Version (KJV)
5 This then is the message which we have heard of him, and declare unto you, that God is light, and in him is no darkness at all.
As God is light, therefore the absence of God is darkness.
1 John 4:8 King James Version (KJV)
He that loveth not knoweth not God; for God is love.
As God is love, therefore the absence of God is hate, suffering and violence.
Philippians 4:7 King James Version (KJV)
7 And the peace of God, which passeth all understanding, shall keep your hearts and minds through Christ Jesus.
Peace comes from God, therefore the absence of God means no peace. There is constant torment in hell.
John 14:6 King James Version (KJV)
6 Jesus saith unto him, I am the way, the truth, and the life: no man cometh unto the Father, but by me.
Life comes from God, therefore the absence of God is death. People in hell experience constant death.
Ezekiel 37:5 King James Version (KJV)
5 Thus saith the Lord God unto these bones; Behold, I will cause breath to enter into you, and ye shall live:
God caused breath to enter man. People in hell are under constant asphyxiation.
John 4:10 King James Version (KJV)
10 Jesus answered and said unto her, If thou knewest the gift of God, and who it is that saith to thee, Give me to drink; thou wouldest have asked of him, and he would have given thee living water.
Living water comes from God, therefore the absence of God means there is no water in hell. People in hell are constantly parched.
Psalm 91:11 King James Version (KJV)
11 For he shall give his angels charge over thee, to keep thee in all thy ways.
People’s guardian angel comes from God, therefore the absence of God means people lose their protection from demonic attacks. In hell, people are under constant torture and attacks by demons.