Yes, I've lived in Seattle without a car, that is easily achievable and many people do this. In the core, you can get around by walking since the weather is mild most of the year, you don't even need to use public transit. You can expand your footprint on light rail and buses, which has been slowly improving its range but many people don't use those if they are in the dense urban areas.

The one gap is that Seattle is surrounded by very popular nature and outdoorsy activities up in the mountains or out on the islands. It is one of the major perks of living in Seattle. If you want to spend a lot of time wandering around the Cascades, you'll need a car. Some people that have a car use it almost exclusively for outdoors activities, not day to day living.