We've stayed in Siena twice, so about 10 days total, once with and once without a car. I strongly suggest staying inside the walls of the town, but using our strategy of staying just barely inside the walls instead of right in the center of town. Quieter at night. Almost always less expensive. So is getting an apartment on an upper floor with no elevator, which also gets you a nice view.
Siena is tiny and a pleasure to walk in. And if you stay near the edge of town, you can be very near to one of the underground parking garages or parking lots --- these may be up on the hill or below the hill but with escalators.
On our 2nd visit, we stayed in this airbnb apartment near the Porta Camollia which is near the escalators that go down to the train and bus station and a parking garage: https://www.airbnb.com/rooms/21406684?guests=1&adults=1&s=2&unique_share_id=1151ce8b-ee87-4496-9ac4-0a7e39351637
Siena is at its best in the early mornings and evenings when the daytrippers are gone. Staying in town, we avoided the mid-day crush of people by eating lunch for 3 hours and by checking out the less popular churches and museums. Or you can go back to your apartment and take a nap so that you can stay up late and eat dinner when Italians do.
A restaurant we like is outside the walls at the bottom of the hill --- it was no joke to walk back up that hill! Only on our last visit to it did we discover that there was an escalator --- it takes you from the public parking lot below the Porta di Fontebranda up to pretty close to the Duomo.
