Yes, boot sector viruses were a very real threat. In fact, up until the mid-90s they were probably the most common type of viruses on home computers.
The earliest types only infected floppy disks. When a computer was booted off an infected floppy, the virus would copy itself into RAM and wait for the unsuspecting user to insert another, uninfected disk, at which point (unless the new disk happened to have the write protect tab engaged) it would copy itself onto the new disk, possibly marking it as bootable even if it wasn't already.
This was actually a pretty effective spreading method, at least for its time. Since a single floppy disk didn't really hold that much data, people were swapping floppies all the time to switch between programs or to access files saved on different disks. And since floppies were also quite prone to failure for any number of reasons, from exposure to heat or magnets to just general wear and tear, people quickly learned to always make backup copies of them. Which, of course, provided an excellent opportunity for a virus to infect the copy.
A major factor the contributed to the success of these viruses was, of course, software piracy. At the time, pirating software commonly meant sharing copied floppy disks with your friends, and if one person happened to have an infected computer, they could easily infect their entire school or workplace. And of course, even perfectly legal sharing of public-domain software (which was also common) could spread viruses just as well.
One feature that aided the spread of these viruses was the fact that some popular home computers at the time (including, notably, the Commodore Amiga) provided a mechanism for background code to survive a warm reboot. The feature was presumably meant to allow software patches to the OS kernel (which was stored in ROM, and thus not easy to update directly) to persist over a reboot, and its description was buried rather deep in the official documentation, but it was soon discovered and enthusiastically adopted by virus writers. In fact, the very first Amiga virus used this feature to infect the next disk an infected computer was booted from after a reboot.
Of course, the virus could not survive a cold boot. But even after this infection vector became widely known, few users would bother to always perform a proper cold boot, which involved physically switching the power supply off (and, if you really wanted to be sure, waiting half a minute or so to make sure the RAM really loses its content), when it was so much easier to just press a few keys (Ctrl and both Amiga keys for the Amiga) to trigger a warm boot.
Later on, as hard disks became more popular, many boot sector viruses acquired the ability to also infect them. For the viruses, the introduction of hard disks was both good and bad. On one hand, hard disks saved users from having to switch between floppies so much, reducing the opportunities for transmission, and booting from the hard disk made it harder for viruses to infect a computer in the first place.
On the other hand, many users would still occasionally boot from floppies e.g. to run specific software that required it, or they might simply forget that they had a floppy in the drive while rebooting (BIOSes at the time being commonly set up to boot from a floppy by default, if one was present). And once a virus did manage to infect the hard disk itself, it would then be loaded on almost every reboot, and thus had ample opportunities to infect new floppies (which people did, of course, still use e.g. for sharing files).
That factors that eventually (mostly) killed off boot sector viruses were threefold:
- The increasing popularity of hard disks during the mid to late 90s meant that fewer and fewer people were using floppy disks for anything except file sharing and installing new software.
- The introduction of CD-ROMs around the same time made floppies obsolete as install media. As CD-ROMs were (originally) read-only, unlike floppies, they could not be infected even if used on an infected system.
- Finally, the popularization of home and office Internet access in the late 90s, and of USB flash drives a bit later, made floppies also obsolete as file sharing media. They also opened up a lot of new and more efficient pathways for viruses to spread, so that virus writers increasingly found the relatively slow (and increasingly ineffective, for the reasons mentioned above) boot sector route no longer worth using.
So, basically, boot sector viruses died out together with their main vector, the floppy disk. To some extent they were replaced by USB AutoRun viruses, which spread in similar ways, but even those could not truly compete in transmission efficiency with e-mail viruses, document-infecting macro viruses and their hybrids.
Of course, the old viruses never really disappeared, and I'm sure that plenty of people still have old infected floppies just sitting and gathering dust somewhere. But very few if any new boot sector viruses are being written any more, and modern antivirus software tends to easily detect any remaining "fossil" infections (if they haven't dropped those ancient signatures from their database, that is). Not to mention that, since custom boot sectors aren't commonly used any more, it's pretty easy for anti-virus software to heuristically detect that something funny is going on if they see a boot sector that doesn't match the few usual patterns.