I have my stuff in a really nice facility on La Brea near Rodeo called Price Self-Storage (3430 S La Brea Ave, Los Angeles, CA - (323) 924-2025). It is a bit out of the way, BUT (1) the rates are good, (2) the facility is really clean and secure, and (3) you can drive your car into the warehouse.
I have stored stuff all over the country and LA and this has been my favorite so far.
Hope you like them too :)
They have a referral program but it requires I come with you, which is ridiculous. Anyways, good luck.
Ryan Ole Hass
The Red Door Group L.A.
Keller Williams Realty
All the best,