I love this community. The amount of good advice you get here is just great.
@FIVEam this really comes down to comfort level. If you aren't confident in your skill level, or ability to follow instruction (which is not meant to be an insult - a lot of the Linux community suuucks at welcoming new people and can be downright mean when basic and previously answered questions come up), I strongly recommend you not commit your treasured family data. Instead, look for cheap alternatives.
The first one that comes to mind is amazons unlimited storage, or back blaze, or any of a bazillion other options. If you have privacy concerns, just encrypt them before upload. This actually isn't a bad practice, anyway, for critical data.
I was able to pick up 128gb flash drives for $15 each recently. That's another option for temporary safekeeping while you expand you knowledge and experience. It's not sexy or very fun, but it's reasonably safe and affordable for your data.
Then, you can begin the process of learning Linux or whatever system you are interested in. You can buy the correct parts for you needs at a pace that fits your budget. It may also allow you to pounce on those once-every-few-years kinds of deals on eBay ...like I did. Or, you can save up for that synology hardware.
You've got some reading to do about your options. Personally, im sold on the used server class hardware and registered RAM route with ZFS. But that's my solution, for my needs.
A lot of what I just wrote sounds like I am dissuading you from going for it, but that's not the intent. The only way you'll figure out what you need is by diving in. So dive in! Build a web server for fun, out of an old pc or on a Virtual Machine. Use a tutorial from digitalocean or right here on the forums. See if it makes sense. Come back with questions.
You're asking good questions already. You just need to get your hands dirty with some d stuff so you can find out if you're ready to make an investment in more stuff.
Sorry for the wordiness.
TL;DR: build a garbage Linux pc and start testing your skills and limit and to see what works before investing.