Closest junkyard to downton LA?

It all depends on what you are looking for, really. If you want classic/stock parts from a 70 'stang, you may have difficulty finding one at a local yard. However, if you are looking for something like a serpentine pulley system, or fuel injection parts, or brakes you may have decent luck.

Here are a couple of places to try:

Memory Lane (they deal in classic automobiles - your best bet for finding stock parts)
(818) 504-3341
11311 Pendleton St
Sun Valley, CA

U-Pick A Part (as the name implies, bring your tools. Best bet for pulleys, brakes etc.)
(323) 875-1105
11409 Penrose St
Sun Valley, CA

EDIT: I'll go ahead and post a link to this thread in the West Coast forum so you may get input from there as well.

Memory Lane in Sun Valley on Pendelton St., they had a few 70 mustangs the last time I was there. Theres also a pick a part next door to it, but I have'nt seen an early mustang there in years though.
Memory Lane - No Tools, They pull the part for you.
Pick a Part - Bring your own tools.

That are a few more junkyards in the area, so just go up and have a look.

For Memory Lane, I'd call them first.

I think there's one on alameda called 'ford only' salvage or junk yard

memory lane and pick yruo part are owned by teh same guy. The 70 mustang will never make it to teh pick your part yard, it would go directly to memory lane. They usually have a nice selection of parts. You can also ask them if they are getting any more cars in and they will tell you if they have one comming in and when.