There are public docks in Marina Del Rey, as well as plenty of docks with public access such as the boat launch. You cannot stay there without paying, but you certainly can land and do a quick pick up. All they can do is tell you to move along.
You can also anchor out, but it's not a great anchorage. But if you want 'long term' you need to get a slip, and it's not easy in MDR or Redondo but further south in San Pedro/long beach it might be more doable.