First thing to say is that your unreliable Vaio may not actually be an issue with regards to you data; it's pretty easy to plug-in a notebook hard drive and read it on any other computer should your PC actually die, although you'll need a simple, cheap adaptor for an older IDE drive to a regular PC (the interface is the same, just smaller, the adaptor is purely to physically connect). Sata is no problemo.
That is unless the hard drive itself is the source of the problem. Saying that though, back-up should be done even on the most healthy of systems as it doesn't stop theft, fire or any other total loss.
I would approach the problem this way. I'd buy a new hard drive for your notebook and an appropriate USB hard disk housing. This will be SATA or IDE for an older machine. it's extremely easy to tell the difference as 2 minutes on google will show.
DO NOT buy a ready-packed drive for reasons that will become clear. Years ago, these were simply notebook drives with the appropriate interface, hard disk manufacturers now ship hard disks with a built in native USB interface, which is no good for what I'm about to suggest.
Using a desktop PC (beg, borrow), clone the disk using Clonezilla (this is free and can boot from a USB stick) to the new hard drive.
Put the new hard drive in the notebook. If it stops crashing, congratulations, you fixed your notebook! Also with Vaios, try replacing the CMOS battery (especially if more than 2-3 years old) and resetting the BIOS to default. If it didn't fix it, then put the old disk in the hard disk enclosure and you have your back-up. You should use the newer disk in your machine as new hard drives are much quicker. Then try re-installing Windows (you have a back-up of your old system remember); most "crashes" are software related but you are right, despite premium prices Vaios actually don't do particularly well in terms of reliability - you want a Thinkpad for that. Macbooks actually have MORE quality problems than many PC brands (screen problems, cracking cases, discolouration of plastics); they are of course internally litte more than generic PCs anyway these days, hence the Hackintosh cult.
What else?
2.5" drives are pretty relaible as external USB drives as they are designed for portable use. I too have had problems with larger 3.5" drives as they are designed for static desktop PCs.
You should also organise your PC. One of the most annoying thing about many PCs "out of the box" is that they use only a single disk partition for Windows. Always, always use a seperate partition for windows (usually C
and move your "user data" to the other partition. This is easy to do by right-clicking "My documents" and selecting a new location. You can now re-install windows on the c: drive without losing anything.
Windows has a built in basic tool for back-up and it works pretty well. Open explorer, right-click the drive and properties->tools and backup now.
Finally, keep your back-up data safe but physical seperate from your computer. The best back-up solution is across the net to a remote location.
Hope this helps.