Lots of questionable advice, here.
First, do not rely on Teh Cloud, in any form. If you don't have physical control over your data, you don't own it, and whoever does own it can do anything they want to it and you've got very little recourse. That could include deleting it, peeking inside it, or even sharing it with the world.
If you're not overly worried about your cloud provider accidentally or intentionally sharing your data with the world, it can make a nice additional supplement to your data archiving strategy, but only as a "if everything else I'm actually relying upon goes tits-up, I'd hopefully still be able to get to it in the cloud" sort of last gasp hope.
With that out of the way, the only reliable method is to continually keep all your data readily available and online and part of your regular backup routine. As others have pointed out, old media die in lots of different ways. If you copy all your old media to whatever you're using today, you don't care if the old stuff dies for whatever reason, and you're also confident that you've got a valid copy. There's no worry that your several-year-old DVDs might be starting to delaminate, or that your Zip drives will have the click-o-death, or whatever.
Yes, that means you need bigger hard drives today, but the good news is that hard drives are dirt cheap compared to whatever you spent on your old media. A single DVD doesn't even store as much as a typical CF card. A hundred bucks gets you a hard drive that holds the equivalent of a few hundred DVDs. When that drive fills up, get another.
The simplest and most reliable backup method these days is to get three times as much disk space as you need. Disk(s) 1 are where you keep everything. Put a copy of everything on disk(s) 2. Every week (or month or whatever), take disk(s) 2 offsite to your bank deposit vault or your parent's place or somewhere you trust and exchange it for disk(s) 3, which you bring back with you and start treating as you used to do with disk(s) 2. The next week, do the swap again.
Also worth investigating, depending on your performance needs and your desire for tidyness, are RAID arrays. Be careful; many commonly-used RAID modes actually put you more at risk for data loss than a single hard drive, meaning you need that much more redundancy in your backups to compensate. Safer RAID modes eat up more disk space. Duh! But you only want to think about RAID if a single disk isn't big and / or fast enough to hold all your stuff, and you should then think of the RAID array as a single disk that happens to have some extra moving parts.
IOSafe also makes near-indestructible hard drives: fireproof, waterproof, crushproof. They're more expensive than a regular hard drive, but very reasonably priced. If you're on a Mac, just get one (or three or however many you need) and point TimeMachine to it (them), and the only reason you'd need an offsite backup is if you're worried about theft.