|Relic skull and reliquary of Saint Ivo of Kermartin (also St. Yves or St. Ives), (1253–1303) in Tréguier, Brittany, France|
1) You don't have to venerate relics (of any class) or have any or view any as a Catholic. So, if it freaks you out, avoid them.
2) Not everything has to have a specific scriptural basis to be a valid and worthy practice. This is the error of sola scirptura which itself has no scriptural basis and is, thus, self refuting.
3) That being said, the practice of relics does have a scriptural basis. The bones of the prophet Elisha bring a dead man back to life in 2 Kings 13, a woman is cured by touching the hem of Jesus' robe in Matthew 9, the sick are cured when the shadow of Peter passes over them in Acts 5, and handkerchiefs that were touched to Paul's body both heal the sick and drive out demons in Acts 19.
4) Remember Catholicism is an incarnational religion - we worship the God who became flesh, i.e. took on matter. Gnosticism (which is present in much Protestant thinking) says that matter is evil and spirit is good. Many today at least believe that matter is indifferent to spiritual things. This isn't a Christian worldview. God created matter (cf. Gen 1), God loves matter. God uses matter to convey spiritual grace. The prime examples of this are the Sacraments, especially the Eucharist, but other "sacramentals" including relics are based on the same reality.
5) Lastly, remember - a relic (or any sacramental) has no power of its own - that is magic and superstition and is clearly taught against by scripture and the Catechism of the Catholic Church. God chooses to work miracles through matter, including through the relics of the saints, but the relics themselves can do nothing apart from the will of God. They merely are the instruments through which God freely chooses to convey His grace.