Hi there.
If you are competent enough to take out a few screws try this.
Normally the bottom of a laptop has a number of panels that can be removed, for RAM memory or hopefully, the hard drive.
If so the little hard drive should be able to be pulled out, usually a plastic tag to tug it out with.
If not, perhaps when you remove the battery you will see this little tag sticking out.
Then beg borrow buy a external hard drive enclosure for a 2.5" hard drive. If this is an 'old' computer, >5 yrs the connection required is "IDE". Noted by a lot of small pins in two rows.
If not a SATA connection required - no pins sticling out of drive.
Plug in drive and attach to a computer, usually USB.
Your old hard drive will just act like a pendrive or similar just a big storage device.
Pull of your personal data onto the computer your working at by drag and frop methods.
If the laptop is 'broken'. Bin it. If drive is working ok and you've got all data off you need. Then reformat it and you have now gained a large portable USB drive for further back ups.
If you don't want to keep drive or it is not working I am told the best way to stop anyone from accessing it is to drill a couple of holes through the drive.
It has been shown that drives put in buckets of water, set fire to, run over by trucks could still be accessed.
Hope this helps.
Cheers