I have a two-step system:
First, I scan almost everything (except minor receipts). Once scanned, I file the searchable PDF on my computer.
Second, if the file is critical (tax-related, medical related, housing related), I keep a hard copy. Every six months or so I go through my hard copies and shred what I no longer need.
I use CrashPlan, an online backup service, to save all my files. It automatically scans my designated folders and as soon as changes are made it backs up the new/modified folders to the cloud. Using their mobile app I can also pull the files on to my phone if need be.
Works like a charm, and I never worry about losing documents.
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I strongly endorse the ScanSnap scanners. I have a smaller, more mobile version that does a brilliant job. The interface is always incredibly easy, which means my less-technically inclined wife has no trouble using it. We also have one of those all-in-one printer/scanners, but the scanning process is much more involved and certainly not one or two clicks as the ScanSnap is.
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I use Evernote for storing and storing book notes, web sites, interesting web articles/blog posts, and for general musings. It's a great piece of software.