Quicken for all the records. Quicken even has a password vault (for stuff you can download, anyway), and you can always save other passwords in the comments area of the account information. Or mint.com, though I’ve tried that and it’s not nearly full-featured enough and it’s a pain to use.
Receipts: if you use binders, then as some have suggested, the binder sleeves (or they even, I think, make plastic envelopes that can close more securely but can be attached to a binder). Periodically, photograph / scan them and trash the originals (use your judgment as to what things you must continue to keep the hard-copies for).
Statements: Scan and store as soft copies (you may even be able to download PDFs).
If you go the file box route, which is an excellent idea: have big manila envelopes or plastic accordion folders as appropriate, one for each account that gets paperwork / requires receipts / has a checkbook. Keep all the folders/envelopes in the big box.
For us: Quicken is a life-saver. I don’t know how anyone with anything more than the very simplest of finances manages without this or something similar.
Any receipt with tax impllcations gets stuffed into a smallish (5x7) manila envelope throughout the year. In January, those get moved to a new full-sized envelope (and year-end tax statements that arrive in Jan/Feb get added to that), while the old manila envelope gets used for the next year.
The “shove it in the envelope” approach works for us: it’s simple, we reconcile it at the end of the year with what we’ve entered into Quicken. For business purposes, that could be your backup for stuff you forgot to enter into the software.
At some point I’m going to scan / shred most of our old paperwork, so it’s more easily accessible. Soon as I have time, that is :P.