We have a couple of laptops that are still technically working, but we have moved on to greener pastures so we no longer need them. Around the corner there's a charity that reworks old tech, so the laptops are heading there. Prior to that I figured it would be a good idea to securely wipe their hard-drives -- by basically writing random garbage to them several times over. Luckily there are quite some tools made for that. After some trial and error here's what I ended up doing:

1) Found an old USB disk, fittingly sporting a Batman logo.
2) Formatted it as a FAT (MS-DOS) drive.
3) Downloaded the free ISO of Darik's "boot and nuke" data wiping tool (DBAN) from http://www.dban.org/
4) Dowloaded Unetbootin from https://unetbootin.github.io/ and launched it.
5) Used this to turn the Batman disk into a bootable disk based on the DBAN ISO.
6) Due to a bug somewhere in this pipeline (Possibly Unetbootin? Or DBAN itself?), I needed to edit the "syslinux.cfg" file in the root folder of the USB disk. This entailed replacing all mentions of "ubninit" with "ISOLINUX.BIN" (and one mention of "ubnkern" with "DBAN.BZI"). Attention, case-sensitivity going on here. (This handy guide helped me to figure this out. It also points out that it is a good idea to remove the "--autonuke" flag in this config file since that leads to a potentially very dangerous auto-nuking USB stick...)
7) Set the BIOS of the target laptops to prioritise booting from USB drives.
8) Plugged Batman into the laptops and them started up - choosing the aptly named "Default" as wipe option.