Searching for “Unicorn” of lodging in Paris for 4

Try Hotel Opera Cadet on Rue Cadet (nice little market street) in the 9th, right off Rue Lafayette. Literally, steps away from Metro Line 7, not far from Gare du Nord. They have quad rooms as well as connecting rooms, 2 good-sized elevators, great breakfast buffet, mini fridges, and modern bathrooms. 4-star hotel may seem to be over your budget, but they have special rates for early bookings and stays of at least 3 or 4 nights. I booked early and my rate for a single for 10 nights was 123€, including breakfast and a 3 day Metro pass.

I found this one on vacationinparis.com. We've used them twice and been very happy both times. The location is absolutely perfect and it has two bedrooms and two bathrooms plus a washer/dryer. However ,it's a bit more than your budget at $310 USD (273 euro) per night. If you do a search here on this forum you'll see there are a lot of good reviews for this U.S. based company.

Good luck!

http://www.vacationinparis.com/listing/245/saint-germain-perfect-two-bedroom

I just looked at Hotel Opera Cadet and they are charging 566€/nt for a double room and 771€/nt for a quad! Crazy!

Are you looking at the right hotel? I just checked and Hotel Opera Cadet has quads starting at 261€ in December.
In any event, prices fluctuate tremendously and the Christmas season is expensive. Hotel Excelsior Latin has family suites that start at 221€ per night but will be more for most of the year. It is located near Luxembourg RER B station and the family suite has a kitchenette with a little fridge. It is one of my go-to suggestions but I fear that if you are traveling near Christmas, it will be above 250€ per night. My family of four stayed in the family room (chambres communicantes) at the Hotel Apollon Montparnasse on Rue de l'Ouest in the 14th for four nights in July 2017. The room is two connecting rooms with 2 bathrooms -- the rooms, like almost every connecting room I have ever had in Paris, are connected by a small common hallway. If the rooms are rented separately, the door to the hall is kept open and the doors to the rooms have separate keys. The rooms are not huge and the hotel is not on the RER B line. We paid 179€ per night for 3 nights and 184€ for one night for one. The rooms had an empty mini-bar that I never used but I think it was a mini-fridge.

A quad room means 4 people in a tiny room and one bathroom. I would look for modestly priced hotels and get two doubles. Connecting rooms are rare. Mini fridges are also not the norm in France although some rooms have minibars but those tend these days to automatically charge when you move stuff around.

You can probably find a two bedroom apartment; two baths are hard even there especially at that price. Most apartments have a washer and some have US style dryers. For only 4 nights, an apartment will be tricky particularly at Christmas where landlords sometimes require rental for the whole Christmas season. Christmas is high high season in Paris for prices and availability for apartments. It will be hard to find someone willing to rent that prime time for only 4 nights. I'd be thinking two doubles in a modest hotel.

Here is a review of our favorite hotel in Paris:

https://community.ricksteves.com/travel-forum/france-reviews/hotel-des-grandes-ecoles-paris-2b604cf6-f7a8-4b38-91fb-d1caefc3c6ae

The Hotel des Grandes Ecoles was around 50 meters from a metro stop, but was also within walking distance of Notre Dame.
When leaving for home, we walked to the Luxembourg RER B stop and took the train to CDG airport.

Our dates aren’t locked in yet, but we’re looking at 12/23-12/27. I’ve had some luck in finding a few unicorns, mostly two separate rooms (some connecting) with minibars. I appreciate everyone's suggestions. Please keep the suggestions coming! I’ll put together a list in hopes of helping others.

hey travel4fun
don't know where the elusive "unicorn" is in paris. we spent lots of time looking for 4 people and a "legal" apartment. found one with TA but no availability for your dates.
look at cross-pollinate.com and put in filters. they are offering "new years" discount of 12% if booked in january
found one with 2bed/1 bath "eiffel tower suite" with alot you're asking for except 1 bathroom.
click book now and this gives you a great discount and less than 600E for your time. if you are very certain book now for a great place and cheap for the holidays. read the fine print and how balance due.
the other is "folie mericourt apartment" in bastille/marais district. 2bed/2bath but has a sofa bed. the total is 800E with discount and read fine print and payment due and how.
it's a busy time during holidays, find a place, book now and one less worry, stress and gray hairs. enjoy
aloha

I stayed at Hotel des Grandes Ecoles for a month, not my favorite. A fair distance uphill from the mêtro.
Try Hotel Diana in the 5th, or Hotel Jeanne d’Arc in the 4th. Both great budget-friendly hotels.
I’d take the 150 sq ft requirement for each room off the requirement list at your budget.

I may get boo’d for saying so, but look at AirBnB. I had to do a LOT of looking but found many legal rentals and eventually booked a great little apartment that I’m very excited about for my family of four — kitchen, bath, sitting room, beds for all. Just narrow your search using the filter function for number of beds, price, desired amenities, and location. Then look at the bottom of the description of each listing — if they are legal there will be a lengthy registration number. If not, move on to the next.

Vacation in Paris is another good website to check out. Email the owner of the company and let her know you are looking for a legal, registered rental and your dates and price range. She will email you back with a listing of available properties.

Good luck!