If you have circles of radii 1/1, 1/2, 1/3, ..., 1/n for all n >= 1, then their total area is finite, although the sum of their radii is infinite.

Can one fit these circles into a finite-sized rectangle?

yes, they will fit inside a 1 x (11/6) rectangle for example. put each circle inside a square, and then pack the squares like this: http://www.stetson.edu/~efriedma/harmonic/ erich