DIY electrical projects can equal a fire, so don't shy away from calling an electrician to make something like this happen if you aren't confident in your electronics/wiring skill. It's smart!
Having said that, this is a pretty easy DIY project. If you get a mid-ranged* charger/controller/inverter (or seperates) at a big box store and pick up an appropriately sized** deep-cycle battery, you have a good start. When the power is out and you need to recharge, you have options to charge the battery at a neighbor's or gas station where the power is not out, or possibly hooking the battery up to solar for complete independence at least when the sun shines.
There's lots of info out there on what you're thinking about doing - a real deep-cycle battery*** battery will be the core of it. Here is a
good link that explains the basics pretty clearly. (Even if you don't add solar, places that deal in solar will be great reasources.)
Also…sorry about your luck!
-Matt
* $50-100 or more would be a good target depending on what you want...cheaper than that isn't going to give you all the options you need
** rated in Amp-Hours....literally how many amps do you need for how many hours.
*** They are made with solid lead plates, designed to discharge down to mostly empty on every duty cycle...marine and car batteries are not the same, but could do in a pinch...durability (# of recharge cycles) will be the difference.