My favorite kind of Sunday is the one where the car stays parked. Coffee somewhere with a table outside, a slow lunch, maybe a bookstore, then a drink before it gets dark, all of it on foot. Where I am you can do about half of that before you give up.
So which streets / neighborhoods actually hold up for a whole day?
Marina del Rey!! Coffee at Blue Bottle in Waterside, walk the whole marina, then Killer Shrimp for dinner. Car never moves.
Koreatown. Everything is downstairs already. You got the restaurants, bars, grocery stores.. everything is pretty much walkable. The density is the point.
You can eat five different things, buy snacks, sit in a sauna, do spa treatments, all while never leaving a 15-minute walking radius.
Inglewood.. once on Market Street and you are set. Coffee and a plate at Stuff I Eat, bars & all the shops right there, like Three Weavers, that’s a short walk for a locally brewed beer. Not fancy, but you never touch the car again.
Larchmont Boulevard, easily. The farmers market on Sunday, coffee at Go Get Em Tiger, lunch at Great White, and three blocks of shops. I park once and forget the car.
West Hollywood, obviously, but the trick is picking one pocket and staying in it. Melrose Place and the design district is mine. Coffee at Alfred, lunch at Gracias Madre, then the shops, all of it inside a few blocks.
Sunday afternoon at The Abbey closes it out, and the walk home takes four minutes. I have done that entire day without starting my car once.
Marina Del Rey would have been the last neighborhood that comes to my mind, but we all have different tastes. 
Santa Monica or Venice are more obvious picks for me. One bar and restaurant next to each other, you got the beach as well. You can see people walking around, people on bikes and scooters, breath the ocean air, enjoy actual grass and sand when you want to sit down. Perfect for a lazy Sunday and a car free day.
The Bungalow is closed on Sundays, but that’s okay, who wants to get hungover before a Monday? 
Old Pasadena is perfect for this. Park somewhere around Colorado and just wander. Coffee, restaurants, shops, bookstores, people watching, random side streets. You can kill 6-7 hours there without even trying.
I like to drive up to Highland Park sometimes. York Blvd for example, especially if you’re not trying to cram a million “activities” into the day.
Coffee, vintage stores, record shops, tacos, bars, just walking around and stopping wherever looks interesting.
Old Pasadena is a great choice for a car free Sunday. 
That’s true, but it’s one hour via car from where I live. Otherwise it’s an obvious choice. So I would go with Santa Monica instead. I barely use my car during weekends. Walkable and chill..
West Hollywood, Santa Monica or Beverly Hills. Streets like Santa Monica Blvd make it easy. The stretch from Santa Monica to Venice offers enough activity for multiple days.
Culver City isn’t that bad either. Lots of things to do.