We were there this past summer. Took my kids and some of our friends on a little Metro Rail train trip to Union Station which is across the street from Olivera Street.
It is, in large part, very touristy. Lots of stalls selling touristy stuff and some things you would find in Mexico. There are a few places to eat. What I found most interesting is that there is a historic home that is perserved as a museum that gives you a view of how they lived when the city was just beginning to form and I think, still under the Mexican goverment. There is also a historic firehouse there with a horse drawn fire engine. Also located just off the plaza is the Chinese-American museum which has displays and exhibits about life for the early Chinese immigrants in Los Angeles as well as an art exhibit by a prominent Chinese artist who did some work for Disney. One stop over from Union Station on the Metro Rail is Chinatown which we also visited.
If you did all of the things I mentioned, it probably wouldn't take anymore than a couple of hours. I think for most it would take less. For most tourist, it's not worth the trip just for that one location from Anahiem. We took it as a little day trip adventure on the train for our children. If you are planning on sightseeing in Los Angeles, say a visit to MOCA (museum of Contemporary Art) or the new Frank Geary designed Disney Concert Hall, then I'd say it's worth a stop. It might even be worth a stop if you're visiting the Hollywood area on the way back to Anahiem. It's right off the freeway. It does give you the flavor of what a little Mexican town might have been like.
I had read in the paper earlier this summer that the city was talking about a big raise in the rent for the merchants there. The merchants said they would have to close up shop if they did so it's possible that in the future, Olvera street would be a largely empty street instead of the little Mexican bazaar that it is now.