Heating and cooling are the biggest energy draws of a household. Living in northern Wisconsin in a straw bale house we don't need an air conditioner. During the winter we use wood or gas to heat with, so that leaves refrigeration as our main concern as far as large energy draws.
We also live off-grid, which means that the only electric power we receive is from our own solar panels, which are tied to a bank of batteries that are filled when the sun is out, and deplete when the sun goes down. Living off-grid requires you to be acutely attuned to the sun and what the clouds have been up to. You wait until it's sunny out to have a disco dance party, or run the washing machine, for example. The problem with refrigeration is that you don't have a choice as to when to keep your food cold. Through some research, trial, and error I've found what works for us.
The Most Efficient Cooling System
In our area and for much of the year the best place to store food is outside. During the colder months of the year we can keep items frozen in a cooler out behind the house. What do you do with that big pot of leftover soup? Don't stick it in your refrigerator. Put it outside to cool first. Wise use of what you already have will save you a lot. During the summer the solar energy we get is more than we can use, so no big deal, but in the winter the sun is out for a shorter period of time, and even then it's often cloudy, so we have to be smarter.
Along these lines, a friend of mine created an indoor cooler with air access to the outside. It is basically an old refrigerator with a hole cut near the top and bottom with PVC pipe vents leading to the outside. The pipes have screening on the ends to keep varmints out. During the winter he can partially close or fully open the vents as needed to have a refrigerator in his house, for zero cost.
The Next Most Efficient Cooling System
Technically, DC powered refrigerators and freezers are the most efficient because you don't have to convert the DC energy (Direct current) from your batteries to AC current (alternating current, like most homes use) Though we have a DC powered chest freezer we use an AC powered fridge (I'll explain why below) so I don't have any experience with the DC powered refrigerators. If you have a very small photo-voltaic system (all the way down to an RV or sailboat) with small cooling needs you might want to check out this article http://www.backwoodshome.com/solar-powered-refrigerators/
Our Refrigeration System
I bought a 10 cubic foot refrigerator from Menards for $350, glued 2" of foam to the outside, and it draws the same amount of electricity as a super efficient $1800+ AC fridge/freezer combo of the same size.
In stock form this fridge used way more energy than comparable appliances. In fact, within a range of 310 to 340 Kwh (Kilowatt hours per year) listed on the yellow tag when I bought it, the fridge actually used 365! After applying the foam to the outside the energy use went down to 251 kwh per year average and thats WITH automatic defrost. (A heating coil that turns on every 8 hours of run time to melt ice buildup) It makes you wonder why these companies don't just put a bit more insulation on their appliances. If applied to all units in the USA it would save millions of dollars in energy costs. But that's another issue...
The key to this setup was to find a bottom fridge/top freezer combo unit with the cooling fins on the back and EXTERNAL to the appliance. It is unusual to find them set up this way now-a-days since most appliances hide the coils within the walls of the insulation! How silly is that!!!
I unscrewed the coils in the back and very carefully nudged a piece of 1" foam between the coils and the back of the fridge. I then glued 2" of foam on the sides and top, and none on the door, and was done. Note that you CAN'T do this retrofit to a fridge with coils in the walls, only the cheap models will work.
If you want a bit more efficient conventional fridge than mine to start with the best I've found is the Danby https://www.danby.com/en-us/products/apartment-size-refrigerators-en-us/dff100c2wdd/ 297 Kwh per year and it looks like you may be able to access the coils from the back as I did with mine. This is the one I'll be getting when ours dies, but who know when that will be? By insulating our fridge it runs less often, and should last longer. We're going on 10 years with this cheap one.
One of the most efficient and cost effective fridge-only options is to purchase a chest freezer and install a temperature control to override the freezer controls. Chest fridges are more efficient for two reasons; They have more insulation, and the cold air stays in when you open the door.
There are many articles and videos online that tell you how to do this and it is a very easy retrofit. Are you ok with the potential access issues of a chest fridge? Do you have the space in your kitchen? If your answer is "yes" then this is a great way to go. Here's a good article about if you're interested https://www.newlifeonahomestead.com/convert-chest-freezer-to-fridge-solar/
The Easiest System
Is to purchase a ready-made high efficiency fridge that the company essentially added the insulation for you. Sundanzer is a good one and middle of the road in price. There are many others.
Hopefully you've gained a bit of inspiration from reading this article. Remember that we came up with a system that works for us and it may or may not work for you. You have to make your own choices, but the research can be a lot of fun!