It should be fairly quiet by midnight on Sunday, the same Monday-Thursday. But since there are no night clubs around the plaza, outside of the Hotel ME's rooftop lounge, I wouldn't worry too much about the noise during the week.

it is a busy spot day and night as are many streets nearby.

When you find an apt. you like, take a look at google (street view) to see
what is around . I find it very helpful.

I haven't stayed at Atocha but I've stayed within 15 minutes nearby at Sol and the Prado and it is a busy area. Plaza San Ana is noisy on the weekends. I would just take the normal precautions that I do in NYC, where I live and be on the alert and not flash money, jewelery and electronic gear about.

Spain is noisy all over. Spaniards abhor silence. As Robert said, you should be OK in that area.

We stayed near Atocha in two separate hotels during our last trip to Andalucia and we had no qualms about walking around the neighborhood. There are many, many little restaurants and bars and the area is full of atmosphere. I don't know if it's a good area for little children (playgrounds, parks etc...) but for two adults, it was great. Being near Atocha means good transport links.