The book seems to cover interesting topics and I read an old review which said the book would be helpful in showing students the concrete side of analysis before delving into the more theoretical side of things. Is this book still relevant today?
I'm currently going through the latter chapters on stability for a grad course. The book has an encyclopedic feel, proves results under fairly general hypotheses and leaves many details (as well as context and motivation) to the reader. The problems are tough but not impossible; some are fun.
It's a good read if you're prepared to spend a lot of time and especially if you have someone who can guide you through it and possibly select the material, or if you can complement it with another book. I wouldn't expect anyone to read and understand it in its entirety, as with most "encyclopedic" references.
Filling in the details, as well as working through the problems is probably a good way to practice the "concrete side of analysis", in the sense of using the inequalities and tricks that are typical in computations, knowing how to use the hypotheses, keeping track of quantifiers ("for each $\varepsilon>0$ there exist $\delta>0$ and $T\geq 0$ such that for $t\geq T$...") etc.
Keep in mind that the book was written in 1955, before e.g. numerical methods were in widespread use to obtain approximations, so the focus will certainly be different from modern books.