I don't think a foundation (such as paper) is REQUIRED to make a string quilt. But a foundation has many benefits: it really helps stabilize the string block so the center strings match up, it serves as a guide for how long to cut each string strip, and it is the easy template for trimming the block once all the strings are sewn on. Without a foundation, I could never have gotten my yellow lattice strips to line up in the picture of my string quilt below. A foundation can be either paper which is removed, or a leave-in foundation which is what I used. The detriment to paper is that it can be time consuming and a little tedious to remove, and the detriment to a leave in foundation such as muslin or Pellon is that it adds thickness, weight and a little stiffness to the finished quilt.
But to answer your question, if you a making a classic simple strip quilt and don't care if any of the strips line up, you can certainly make one without a foundation.
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