Adam and Martha's place is a great dog friendly home if you are looking to explore the city via light rail. It is a 10 minute walk to restaurants and the library in Beacon Hill, a mile away from Blue Dog Pond park, and also near I-5 if you're planning to drive.
We stayed at Adam and Martha's place for a month, and found it to be a great value. It is an older home, but as long as you understand that, you'll be comfortable (like washing dishes by hand). Adam and Martha have stocked the home with essentials and there was plenty of soap, towels, and laundry detergent. The beds are a box spring/mattress/pillow top combo, which may be uncomfortable after a while for some.
Adam and Martha are very quick to respond to any messages and were helpful with anything we needed. I would highly recommend this place if you're looking for a dog friendly home.