Whilst a laptop screen comes with the machine, like a desktop, it may be required in order for you to see what you are doing (locally), but, it is not actually a critical system component that will make the machine fail (such as having no/bad memory or processor etc.).
If this laptop is covered, you may want to check either your warranty or household insurance to see if you have accidental damage cover... Many people do without realising. If you do not, a replacement screen is usually around £50-£100 (depending on spec/size) and if you are comfortable with opening desktops, you may be able to do this yourself.
Whilst you are sensible about leaving laptops alone, as long as you have a careful hand, with the vast majority, it is quite easy to replace components.
You do not need the screen in order to access items on your hard drive. The first and easiest way, if your computer supports it is to try and use a cable from your laptop to either a spare computer screen (if you have one) or a TV screen. You can get converters between VGA/HDMI/DVI very cheap now and if your TV has one of these ports (most flat screens do), you should be able to display your laptop out on it.
Many laptops will automatically mirror the image, but, you may want to see if you have a dedicated button that needs pressing:
That being said, if you really are not concerned about the laptop, you can always take out the hard drive (with most, they are very easy to get to) and either connect it directly to a desktop (if it is regular SATA) or if it is a laptop/you don't want to open up another machine, you can get a USB dock and get your files from it.
(usually a square type cover, or has a picture of a hard drive near by)
If you do go to putting your hard drive inside another machine and have been using security, you may find this interesting
*, as you will have to override NTFS security on the drive.
(Unless it is Micro SATA, You just need a dock/housing that supports 2.5" SATA or IDE depending on drive. If you are technical/build machines as you said, You may like buying a proper dock such as in the picture as it is always handy to have around).
* (Just did a quick search here, you need to basically take ownership).