This is a very good book for a math enthusiast of any level. I found myself constantly fascinated by the problems Stewart chose to include, and on more than one occasion I made a note of a math problem to think about later. In short, the problems (or perhaps more accurately, problem situations) discussed in this book are very high quality.
As a college-level mathematician, I can't say for sure that the book is accessible. However, I believe it to be so, because the explanations and pace of the book seemed quite slow to me, and because most of the chapters did not presume any prerequisites as far as I recall.
Each chapter in the book is a different (mathematical) subject of study. Most of the subjects are introduced by stories involving various entertaining characters and dialogues. In general the stories do not take away from the math, but they do bring the pace down quite a bit.
After reading this book I have added Ian Stewart to my list of math authors to read. I would recommend the book to most, if not all, of those interested in recreational mathematics.