What is the most central area to rent an apartment in Pais?

The 1st, 4th, 5th, 6th, and 7th arrondissement are the most central locations in the city. Each borders the Seine. Look for a place within a few minutes walk to a Metro and a bus stop. Google maps will help with that. The usual tourist sites of Paris are quite spread out, so easy access to a Metro, preferably a station with more than 1 line, will be an asset.

hey hey johnawhittington
you title should say Paris.
what does "without breaking bank" mean? you have 5 adult people with is not an easy task when during high season, plus the many days you're staying. what is your budget in euros and only staying in Paris?
what is you amenities you're hoping for: A/C, no mobility issues, elevator/lift, many stairs/steps if higher floor to carry luggage up, your gang can carry their own bags, washer/dryer, cancellation policies, beds wanted looking for "real" beds not a small sofa, needing 2 bathrooms,
you are late getting started with many places booked many months ago. it will be crowded so just go with the flow.
look at booking.com, tripadvisor for vacation rentals, parisperfect.com, parisvacationapartments.com. vrbo/airbnb.com. many apartments with them have been cut back with the laws that have happened in the past few years.
hope you look for a place before buying airline tickets. good luck in apartment hunting or look for 2 hotel rooms (3 in one room 2 in another)
aloha

As a family of 4, I wish you much luck. It will be difficult, and expensive, to find a centrally located apartment that sleeps 5 on real beds. We tried and tried for our family, and ended up getting an “apartment” at Hotel Residences des Artes. It’s actually two rooms next to each other. A jr suite with sitting room with sofabed, kitchenette, huge bathroom, and huge king bed. Palatial space by Paris’s standards. Next door is a room with two twins, bathroom, mini fridge. Decent size for Paris. Perfect location right across the Seine from Notre Dame.

I can also recommend Citadines St. Germain. They may have an “apartment” or two rooms that might work.

Arrondissements 1-7 are central but I agree that finding an apartment that sleeps 5 is like finding a needle in a haystack. It will be easier to just get two connected hotel rooms.

The main sights are scattered, you can stay anywhere in Paris and enjoy your visit. There are métro/bus connections to anywhere in the city and nowhere is much more than a 30 minute métro ride away. There is no real compelling need to be in any specific location, as long as you are in Paris.

The only way to offer real assistance is to know your budget window per night, preferably in euros, and how many rooms do you need. You might want to consider Citadine or Adagio apartment hotels.

I do agree you need to give a budget, but I will give you a company that rents apartments in the center of Paris

Guest apartment services rents apartments on the islands in the center of Paris on the islands. They are not the cheapest place in the world, but I generally find their apartments are comparable to a hotel room in terms of price however they only let you check in on Sunday or Monday as a general role which may not work for everyone. The service is good. There is an office with a person there who works as kind of a concierge.

The only thing I'd add to the above posts is that it's very, very late to be looking for an apartment to book in Paris, and especially for a group that large. Savvy travelers know that vacation apartments in Paris are ideally booked at least six months out, and often a year in advance.

It will likely be very expensive if you can even find something available. First thing I'd do is try the apart'hotels such as Citadines and Adiago. I'd also have a look at Le Jardin de Verre, because it's brand new and relatively unknown. It gets great reviews.

https://www.lockeliving.com/en/paris/le-jardin-de-verre-by-locke

Please make your self aware of restrictions and laws regarding apartment rentals in Paris now.
You don’t want to have a place cancelled or not available when you arrive.

I would avoid Airbnb and vrbo type places and only rent through agencies that manage groups of apartments, or look at apartments/hotel type places.

There have been a few discussions on here about different agencies.