If your post had started with "
Very limited power needs. Lights, device charging and some outlets for this and that", then I would say yes. Powering a refrigerator though is NOT a limited power load. Expect a refrigerator to consume at least 1kWh per day, for a small frig. A more deluxe two-door refrigerator/freezer will be more in the 2kWh per day range.
You don't really want to use the term "solar generator" here. This term mostly refers to little wheeled cart systems that you roll in place to supply power to something you're working on outside. What you should be asking about is an off-grid power system.
Since this question comes up over and over, and I have the numbers for my own off-grid cabin, I can throw out some specs for you. The minimal values you can expect would be 2-3 kWh of consumption per day. I'd say you could throw together a workable system with a 200+Ah battery bank at 24V, 1000W of panels (4 250W grid-ties), a 50A MPPT charge controller (look at Epever's 50A), and a 3000-4000W sine-wave inverter.
High-frequency all-in-one inverters typically have next to zero meaningful surge, so I would not recommend you buy anything HF unless someone else here has confirmed the unit you want to buy has ran their refrigerator. Frankly, I see more posts here on what these inverters can't power instead of what they have powered.
Here is a quality inverter that I would recommend. It's what I use in my workshop's 24V system.
Free Shipping! Schneider Electric Conext SW4024-120/240 Inverter/Charger, 4000W, 120/240VAC, 50/60Hz, 30A Transfer, 24VDC, 90A Charger, RNW8654024
ressupply.com
It handles the start-up surges of all my power tools, and it's a split-phase 120/240V inverter designed to be wired directly into your cabin's main electrical panel. It also has a dedicated generator charging circuit. You wire a generator plug circuit on the outside wall of your cabin, then you can charge the batteries directly just running the generator's power through the inverter. No transfer switch required. The inverter itself is the transfer switch. Here's another less expensive inverter, that can likely run your refrigerator.
https://ressupply.com/inverters/samlex-pst-1500-24-pure-sine-wave-inverter
Here's a minimalist system that might get you started.
4 200+Ah 6volt golf-cart batteries
4 250W grid-tie panels
50A Epever charge controller
1500W sine-wave inverter
5000W generator
You could put this together yourself for about 2000$.