This is such a complicated topic, since there are so many variables.

u/whaletacochamp has hit on a lot of great topics to think about.

I'm no expert by any means, but I take a key interest in energy (I'm a physicist, so it is a thing I think about a lot, I guess).

You managed 4 days without power, so I'm pretty sure you now have a good idea of what your needs are for the household, so make a list of priorities and for how long it would need to be an option.

Data is your friend here. First, track your normal energy usage globally; look at the use in the house over a year, but best to log it over a few years to account for variations. Pretty easy to do with your electric billing. Next, look at energy requirements for the things that are important to make the house function, for example, Refrigerator, well and sump pump, heat sources, basic lights. Some can be measured directly with a Kill-A-Watt meter in an outlet, others might take some math (like calculating several 15W LED's on a light circuit).

I'll reiterate u/whaletacochamp here, you NEED a transfer switch to prevent power flow when you switch between an islanding power system versus the power grid.

Moving on...

Weigh the Pros/Cons of every system (I'll list a few):

Battery Backup*:

Pros-

Cost: Can range from pretty cheap cost upfront (but there may be hidden costs for a turn-key system from a place like GMP), to very costly upfront (and later, if not well-designed).

Maintenance: Low/No Maintenance.

Readiness: Always ready/generally seamless back up (Charged up via grid before failure**, or with solar/wind to add to the system post-failure).

Lifetime: This is a wildcard, as the systems are pretty new, but claim to have a 15-20 year lifetime.

*This highly depends on how you set your battery up and any agreements you have with the electric grid provider. Lots of research is required to have a fully independent battery system, but the grid provider cannot touch your 'saved' energy.

**There is always the possibility, although small, that the grid is using your battery prior to grid failure, so it's not guaranteed it will be fully operational at any given time.

Neutral -

Power: Scalability costs money, and there are less installers with experience.

Cons-

Depending on the battery distribution scheme, your household power consumption may mean it lasts less than 24-48 hours. If you only power key sub-systems, or are willing to shutdown breakers, this might go further; especially if you can recharge the battery via solar or wind).

It's a black-box system, unless you build it yourself, or like to hack (ymmv, caveat emptor).

Standalone Gas Generator:

I'd only recommend small gas generators in an emergency, but a small contractor-type generator that can do 3000W-5000W is a nice thing to have. Have several heavy-duty power cords to plug in and a place to set it outside the house while it runs, you can run a few things in the house directly. I have a small Honda 4000W that can easily run at nominal load for 24 hours on a tank of gas.

Use ethanol free gas, start it twice a year to make sure it runs well. Change the oil once a year. Keep a clean run of ethanol free gas in a 2 gallon can and it's all good for an emergency situation for a few days. This is a great option for less than $1000.

Full House LPG/Diesel/NG:

These are expensive to install, but can easily be sized to run your whole house as long as there is fuel. A smaller scale system isn't that much cheaper to build.

If you have the infrastructure to supply them with fuel types, it's not a bad thing to think about, especially LPG or NG.

Pros-

Power: Easy to scale to full house requirements, smaller range between minimum and maximum power costs. Lots of installers out there with experience in setting these up.

Proven reliability/lifetime with maintenance: Over a century old technology, just more bells and whistles.

Cons-

Cost: Really steep if there is no fuel on-site, plus maintenance costs. Reliant on fuel costs/supply.

Maintenance: Minimum of annual professional maintenance to check engine/generator. Can be cheaper if you know a thing or two about engines or generators. Same goes for repair costs.