I wholeheartedly agree with
sheetz about eating at the Original Farmers Market (at Third Street and Fairfax). There is a great array of food -- both table service and counter service -- spanning all different sorts of cuisines. You want gumbo? They've got it. You want BBQ? They've got it. You want freshly made English toffee? They're got it. You want French crepes? Yep -- they've got that too. Many of the merchants have been there for many decades and have kept the businesses in the family, so they have perfected their recipes and kept them consistent. Charlie's (family-run business, and the lady that owns it used to work there when she was a teenager) is just an unassuming little counter service place that doesn't look like anything special, but it was named in Travel Channel's
Burger Land as being one of the places to get a great burger in Los Angeles.
There are also quite a few newer establishments at the Farmers Market as well -- Dylan's Candy Bar, Marmalade Café, Pinkberry, Ulysses Voyage, Ludwig Biergarten, Starbucks, etc.
Here is a link to the Farmers Market merchant page :
http://www.farmersmarketla.com/merchant
So there is truly something for everyone in terms of food and drink at the Original Farmers Market. Plus, The Grove (a great outdoor shopping center that is immediately next door to the Market) has lots of interesting dining options as well.
The Grove's dining options -
http://www.thegrovela.com/dine.php
In fact, I was just at the Market 2 days ago and I couldn't decide what to eat! I had a gift certificate and I could not choose what to eat, as there were so many options. I kept going back and forth between Bryan's Pit BBQ, Patsy's Pizza (I wanted a meatball sub) and a giant fruit salad sold at one fruit stands, and I finally gave up. The choice was too difficult to make.
As sheetz said, the Original Farmers Market is very close to the Tar Pits. The Market is on Third and Fairfax. One way (but not the only way) to get to the Tar Pits would be to leave the Market, head south on Fairfax a few blocks, then turn left (heading east) on Wilshire. The Tar Pits (which is right next to the Los Angeles County Museum of Art) would be on the left (north) side of the street, just a few blocks up. You could also take Fairfax south and turn left (going east) on 6th Street instead of Wilshire, which would mean that the Tar Pits would be on the right (south) side of the street).
It is very walkable if you know the area. I would certainly walk it because I live in the area, and I have occasionally encountered tourists who decided to walk from the Market to LACMA, and I showed them which way to go. But you may feel more comfortable taking a taxi (the taxi ride would be short, so not expensive at all).
You could also try Canter's Deli on Fairfax, which is an iconic Los Angeles eatery (Bono was just spotted there a couple of weeks ago) located slightly north of the Original Farmers Market.
