I looked for weeks a couple years ago to find a low-cost car storage location in Seattle. It was hard.
If you're talking about a week or two, heck with it - there are plenty of street parking spots in Ballard and elsewhere that are adequate for a week or two.
If you want to pay multiple hundreds of dollars per month, there are a small number of indoor garage/storage options around town. They are pretty easy to find in the Yellow pages under "Auto Storage" or something similar.
However, if you're willing to let your car sit out under the elements and you need to leave it somewhere for multiple months, we price shopped and the best we could find was a ShurGard long term storage units and parking spots place out near SeaTac airport. It was decently secure -- although I wouldn't leave anything inside. I think it came out to $70/month or so. The other benefit of this place is that it's close enough to the airport that you can just take a cab from there 5 minutes to the airport and head out... and the reverse when you return.
There may be a few other long term parking places out near the airport as well.