In Christianity our experience of peace is inseparable from our relationship with Christ. It's through God that we experience this deep peace in the first place.
"Peace I leave with you; my peace I give you. I do not give to you as the world gives. Do not let your hearts be troubled and do not be afraid," John 14:27.
Here Christ tells his disciples that through him they have peace, and that the peace he gives is in every sense out of this world. This course to peace is reiterated in John 16:33 when Christ says: "I have told you these things, so that in me you may have peace. In this world you will have trouble. But take heart! I have overcome the world."
Again he proclaims that our route to peace is through him, and he also alludes to eternal peace as he does not promise that our lives on earth will be free from turmoil but that beyond our world we will experience the peace he offers in a much greater sense.
One of the common uses for the term "at peace" is to refer to someone who has passed away. In the sense that it's used here it doesn't simply refer to the removal of earthly worries but it also speaks to the promise of heavenly peace, and we find further links between death and peace, and the detail of eternal peace in the Bible.
We read in Isaiah 11:6, "The wolf will live with the lamb, the leopard will lie down with the goat, the calf and the lion and the yearling together; and a little child will lead them."
The pairing of these animals, predator with prey, symbolises an end to the existing way of life and within the context of this chapter, an eternal peace which accompanies Christ's coming. Whether you interpret this arrangement of the animals in a literal or metaphoric sense, the meaning remains the same - through Christ we have access to everlasting peace.
Another illustration of eternal peace can be found in Romans 14:17: "For the kingdom of God is not a matter of eating and drinking, but of righteousness, peace and joy in the Holy Spirit".
As the kingdom of God is being discussed here, we can know that the peace is being referred to in an infinite sense. Furthermore, as this peace is heavenly it's more than just a lack of something negative but the state of being in God's presence and being filled with this divine emotion.