I've been getting this problem, it started shortly after the Verizon update for my phone; of course this might not be the problem, but it is a coincidence.
Even though I get "Unfortunately Google Contacts has stopped"; clicking on the message, and waiting a short time will make it available again. However, some time later, the message re-appears. I'm looking for solution also. Droid Razr HD - Verizon.
What is getting really irritating is when I can't answer the incoming call because I have to click OK on the error message before I can answer the phone. It is not always quick to close, and I've lost calls. In reality, I think it might be possible to say that this problem started after Google sold Motorola. Perhaps it is happening because the new owners are not honoring our privacy agreement we had with Google.
I found a solution to the problem:
Open "People".. Tap on the 3 dots.. Then "Contacts to display".. Choose "All contacts" instead of Google only.
Now check whether the problem persists.
You'd need to be more specific; when you say Open People, what exactly do you mean; I have nothing in my phone as an app or section, that I can find, that is called "People" so, what do you mean Open "People"? Could you list the steps to get there from the main screen. What APP to open... etc.