If you find yourself in the path of a natural disaster, your first responsibility is protecting lives. Stuff can be replaced. People cannot.
That said, if you have time and can do so safely, there are some things you should do in advance of, for example an oncoming category 4 hurricane, to keep your financial life in order.
Protect your documents
Take digital pictures of any documents you may need after a storm or other natural disaster -- insurance policies, passports, birth certificates, and health insurance cards, etc. If possible, you will also want to electronically replicate financial records, medical information, property records, and estate planning documents (like a will or living will).
Store the originals in waterproof bags in locations that are less likely to flood. Or consider a safe deposit box at the bank. There are also free apps on both iOS and Android that make it easy to use your phone to scan text-heavy documents and turn them into PDFs.
Take "before" pictures
When you file an insurance claim, your carrier will generally only pay out if the damage comes from a single, covered event. Sometimes that's obvious, but sometimes it's less clear.
Taking pictures of your home and property beforehand can make it easier to show that whatever damage occurred was done by the storm, and was not a result of longer-term wear and tear (which is not covered in most cases). Take pictures of rooms from multiple angles. Open your closets, drawers, and cupboards, and snap some pictures of the contents. If items wash away or are otherwise destroyed, those images will prove that you did in fact possess them.
Have cash on hand
After a natural disaster, there can be delays before electrical and financial systems return to normal. Some businesses may open (perhaps on a limited basis), but they may not be able to accept plastic due to an inability to connect to the credit card networks. ATMs may be down, too. So make sure you have a reasonably large stash of paper money in your wallet in case you get the opportunity to safely leave the house after the storm has passed.
Move your stuff
If you live someplace prone to flooding, consider moving your car and other valuables to higher ground. In many cases, city garages offer free parking during major storms, and, in others, it's worth it to pay for the privilege. If you live in a multistory home, consider relocating your valuables to secure locations on higher floors -- or at least get them as far as possible from ground level when possible.
Pay your bills
After a storm, the internet may be down, and cellphone networks' bandwidth gets overtaxed. Mail service may also be stopped or delayed.
To prevent that from causing you any problems, pay your bills, or set them to autopay, before the storm strikes. Doing so can help you avoid late fees and give you a bit more peace of mind. It will also save you from having to try to log in to various businesses payment sites remotely during a time when the cellphone network -- and you -- are stressed.
In addition to taking the steps above, you should follow all local government directions for protecting your property. It's too late to install hurricane shutters, but it's not too late to board up windows and glass doors, if that's what the authorities recommend. You may also be told to use sandbags to ward off flooding if you live in a low-lying coastal area.
Put physical safety first. But if you have time, these tactics will help you protect your finances, too, and make whatever damage a disaster might do easier to deal with afterward.