You do not need a large telescope to observe double stars. A 3 inch (75 mm) refractor can produce excellent results, since its narrow field renders high-contrast, pinpoint star images. A 6 inch (150 mm) scope, though, is better for resolving close doubles.
You can observe double stars from the suburbs and even during a bright Moon. Because stars are point sources, ambient light generally does not overwhelm them.
Double-star observing is enjoyable on many levels. Some observers delight in aiming their telescopes at random stars and looking for companions, or they might try to "split" close pairs.
Others take a more serious bent—measuring, sketching, and photographing double stars, and then contributing their observations to a group such as the Double Star Section of the Webb Society. Such organizations particularly need help monitoring long-period binaries; with separations of at least 0.5 arcseconds, these are the binary type most suited to small telescopes. There also exists a great opportunity for amateurs with CCD cameras to make a significant contribution. There are thousands of wide faint pairs that have been unobserved for years.