That's actually the place to start: your risk.
Everybody has a different risk profile, and you should do some thinking about yours. Think about the types of data that you have on your computers, phones, and various devices:
- Each of these has an operating system, which actually takes up quite a lot of disk space. However, as long as you can re-install (say, from a recovery partition), you probably don't have much risk here. Quick way to check: if your device has a "factory reset" or equivalent sort of feature, you're probably good to go. Personally, I wouldn't back this up because resetting will cause the system to download all the updates since it was installed. Importance level: LOW.
- You will have some sort of documents - letters, budget spreadsheets, taxes, that sort of thing. Recreating this will be a royal pain in the tail end, so you will want to save this. It's also probably not much data - you won't need a lot of space for this. IMPORTANT NOTE: if this data is for a business that you run than this is MUCH more important. Importance level: HIGH.
- Photographs are often times irreplaceable. They also take up a lot more room than other documents. Importance level: VERY HIGH.
- Music (and video) files are the biggest consumer of disk space - the rule of thumb is that MP3 music requires 1 Mbyte of disk space for every minute of music. That's 1 GB of disk for 20 albums. If you are like me and buy music on CDs (I LOVE used CD stores) than you already have a backup for this. If you like to buy music from iTunes, then if you burn it to CD you will also have this backup. Otherwise, your backup needs will get much bigger. Importance level: NONE (music on CD) or HIGH (iTunes).
So for the Borepatch household, our backup needs are modest - a fairly small size for documents and a larger one for photographs. None at all for OS or music. As an example, the laptop I'm writing this port on only has a couple Gigabytes of music (I haven't loaded much) but almost 8 GB of photos.
Note that your mileage absolutely can vary, perhaps by a lot.
Also, and this is important - you need to calculate this for every computer and phone you have. You will have some homework to do before you can start even thinking about what device to get to back the data up. I mean, how frustrating would it be to buy a shiny new storage thingy and find out that it's too small?
Remember, too, that the number of devices that you have will grow over time. You will need more room on your backup device in the future than you need today, just because you have more devices. You will also need more room because you will accumulate more data over time. Since I don't want to replace my backup device very often, I like to have maybe ten times as much data as I think I need.
For Castle Borepatch, all the devices add up to probably 200 GB - almost all of this is music and photos. This means that I need 2 Terabytes of backup space. It sounds like a lot, but it really isn't. Here's one that you can get for a little over $150:
It's a Western Digital WD MyCloud Personal which comes in 2TB, 3TB, and 4TB versions. It will plug into your home router so all your computers will be able to connect to it over WiFi. There are other choices as well, lots of them good.
So you've calculated how much backup space you need that will give you sufficient future growth room. You've found a decent device for not too much money. This will give you one backup location. If your house burns down, you will lose not only the computers with their data but the backup device as well. Part II of this series will discuss off-site backup to address this.
Part III will discuss what software you can get to make backing up your data automatic. If you have to remember to back up, you will forget, and that's when you lose data.
UPDATE 9 January 2018 10:51: I can no longer recommend Western Digital. However, there are a number of vendors selling similar products.