There are some places that have a reputation that you can easily meet someone of the appropriate sex there to start dating. This is usually a bar or a club. But sometimes, there are some strange places (grocery stores, coffee shops, etc.) that have a similar reputation - people who go there are looking for dates.
Those places are sometimes slangily called "meat markets." (this has a bit of a disparaging tone).
So you can have conversations like:
"Is that what you're wearing to church?" "Well, it's kind of a meat market these days.... so..."
or
"I don't want to go to that bar. It's such a meat market, and I just want to talk to my friends without being bothered."
or
"Why is he going to the grocery store across town?" "He's been single for a while, and it turns out, the store on 10th and Main is a total meat market."
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Robin is saying that she's tired of going to places where the focus is trying to meet men.