The inverter I'm looking at can do 3kW continuous, 4.5kW (150%) for 10 seconds and 6kW (200%) for 2 seconds. It's not that I need so much power at any given time but I don't want to run the inverter too hard. There is no single tool I own and that runs from 230Vac that could fully load the inverter, but I will run combinations of tools like the table saw (rated 2100W) and the vacuum cleaner for dust extraction (1400W). The vacuum needs 1400W all the time while the table saw is rated for 2100W and actual power will depend on the load obviously. Running idle it needs about 800W and it will hardly exceed this value during startup because of the soft start. But in theory this combo is rated for 3500W (and not by coincidence as it is designed to run on a 230Vac circuit with a 16A breaker = 3680W). The saw will only go this high when, for example, cutting through thick hard wood so I'll very likely never exceed the inverters maximum continuous power output, even on peaks.
Weight is an issue for me as well. Obviously I want it to be as portable as possible with as little effort as possible. The problem is that while you probably won't need very much actual energy you do need the high currents. Taking my table saw and vacuum combo as an example, they only run for seconds or a minute or two to make some cuts but in that short time the power is needed. Therefore you need lots of cells, or a few big cells, and you automatically get the bigger capacity and therefore energy. For example, for 3000W AC load you need 3300W DC input to the inverter and at 24V that comes down to 137.5A which is a lot.
I thought about LiFePO cells (between 60Ah and 100Ah). With a 3C discharge I'd have between 180A and 300A which is my target zone for this application. Obviously as high as possible without making it too big and heavy. But definitely not under 50Ah so I'll have 24Vx50Ah = 1.2kWh at least which would be enough to run my setup for at least 20 minutes which is much more than I ever need but I have as much energy because I can't use smaller cells. In the end I will likely end up closer to 100Ah so that gives me 2.4kWh just because I don't want to stress the cells too much. I never had to think about runtime for this project, it's solved by itself. I even thought about RC LiPo batteries because they can handle the current and are very light.
But you might need less. That's why it's important to think about the tools first. Many tools will need less. Most power drills, saws, small angle grinders, sanders and so on will need (much) less than 1kW.
A fridge is a much smaller load. Except you have a huge and old one which is really inefficient. My fridge/freezer combination uses no more than 60W (just if it's very warm outside) and usually more like 30W. It's roughly 0.5kWh per day. A battery backup for it would be very small and light unless I need several days / weeks of runtime.