You have gotten several good ideas here, basically though you have to decide how much power you really need and how much you are willing to spend or how much you hate hearing a generator run for hours. The Honda inverter ones are quieter, but I still feel a sigh of relief and peace and quiet when one is shut off.
Last year I set up a system for my parents off grid camp. Propane heat, stove, and fridge. Lights are 120V electric, NO 12V lights and there is no running water. The camp is built basically as if it were an RV. It has a breaker panel with a couple circuits. It has outlets on every wall (per code) and on kitchen counters and lights in each room. The light fixtures are standard Edison base light bulb fixtures, chandeliers, etc. however the bulbs in them are 120Vac LED bulbs. Just as in an RV there is a bank of "house" batteries connected to an inverter / charger. The inverter / charger is typical of a RV type where there is also a 120V input to plug in your RV cord into the "park power". The camp has a power inlet like you would have on your house for a portable backup generator. When a generator is plugged in, the inverter / charger bypasses the inverter and runs the lights and outlets on the generator. At the same time it also charges the batteries.
In addition to the inverter / charger, there is also a set of solar panels totaling 200W connected to a MPPT charge controller, and then to the batteries. With their normal use, they rarely use the generator.
Your largest total energy consumer is that toaster, but I understand why you want it because I've tried to make toast on a stove before. The TV is the second largest energy consumer. I sized the inverter at 2,000W, and the generator is just an inexpensive portable 3500W 120V generator. If they want to use a coffee pot or toaster in the morning, they start the generator, then shut it off as soon as breakfast is done. Actually with just a couple 12V 90AH batteries, they can easily make a pot of coffee and a couple slices of toast, just on battery power. However, that might leave them, later that evening without enough power for lights for the evening's card games, festivities, etc. If I were in your situation, I'd spring for larger batteries and or a larger solar array. Solar is coming down in cost quite rapidly. Also, if you want to place the batteries inside your living space, I would recommend getting sealed type batteries, and NOT the flooded or "maintenance free" batteries. Sealed type batteries cost a bit more but don't give off toxic smelly gas when charged and also don't corrode as much.
Learn to do energy calcs. For example, a 600W toaster operated for 0.25 hours consumes 600*0.25 = 150WH (Watt Hours). From a 12V battery that would be 150WH / 12V = 12.5AH (Amp Hours) So you could be running your toaster for 15 minutes or a 12.5A load from the 12V battery for 1 hour, and use the same amount of energy. Running the toaster for a total of 1 hour would completely deplete a 50AH 12V battery. Now all these are assuming 100% efficiency inverter, etc, so there will always be losses and safety factors, etc... If running 120V incandescent bulbs, take a look at the LED replacements, you can run 6 LED bulbs for the same amount of energy as 1 incandescent. Or another way, LED bulbs give you 6 times the run time on the same battery.
Even if you don't want lots of solar power, I would still recommend a small solar panel to keep the batteries charged and from self discharging if you plan on being away for months on end.
