Most of what we know about St. Anthony is from a biography written by St. Athanasius. According to St. Athanasius, when St. Anthony was about twenty he walked into church and heard the Gospel parable of the rich man who was told by Christ: If you want to be perfect, go and sell all you have and give the money to the poor....Then, come follow me.
In order to devote his entire life to Christ, St. Anthony went out to desert with the intention of spending the rest of his life in solitude and prayer. His reputation for holiness soon attracted followers (so he didn't exactly live in complete solitude). This was the beginning of monasticism, and thus it was also the primitive foundation of religious life as we know it today.
It is still possible to be a hermit this day and age. There are contemplative religious orders (such as the Carmelites) who have an eremitic spirituality, and other orders (like the Carthusians) whose members actually spend the majority of their waking hours alone with God. The 1983 Code of Canon Law, in addition to restoring the ancient order of virgins, also makes provisions for non-religious hermit in can. 603.
In other words, if you felt called to be a hermit like St. Anthony, you could figure out where and how you would live this life, and then ask for your local bishop's permission and blessing. If the bishop supported you in this endeavor, you would be and "official" hermit in the eyes of the Church. Not many people chose this path, for practical reasons as much as spiritual ones, but it does remain an option!
While I personally have never felt called to be a hermit (although for some reason a couple of priests have suggested that I seriously consider it), I do appreciate the eremitic vocation. I find the concept of constant prayer and total dedication to God very attractive from my own subjective point of view. And objectively, I think it is important that we have people like this in our world today.
I think the charism of St. Anthony's life is very well encapsulated in his prayer from the Breviary:
Father, you called Saint Anthony to renounce the world and serve you in the solitude of the desert. By his prayers and example, may we learn to deny ourselves and to love you above all things. We ask this through our lord Jesus Christ, your Son, who lives and reigns with you and the Holy Spirit, one God, for ever and ever.