It depends. In theory, a defective disk should not be able to ever take the operating system or host down (unless the defective disk is the root disk, and the root file system vanishes, at which point the system can not go on). In practice, everything n the planet, and this goal is not reached.
On high-quality enterprise grade SCSI (that is, SAS) hardware, for example LSI Logic = Broadcom HBAs, disks can nearly always fail in all manner of bizarre ways, and the OS itself will stay up. Only very very rarely does a bad disk inhibit the system from working. One example was a disk that was not functioning in hardware (platter, actuator), but was communicating over SAS, and it was able to hold up booting (before the OS comes up) for about 2-3 hours, because the boot loader stupidly retries reading from the disk "forever". Another example was a disk that misbehaved in such a funny fashion that it confused the SAS expander (sort of a SAS switch in a big disk enclosure) so much that the expander had to be power-cycled, and until then about 60 disks were inaccessible, at which point it became pointless to operate the system at all.
With SATA, the situation is different: I've multiple times seen a whole motherboard going catatonic when a SATA disk that is directly connected to the motherboard becomes faulty, or has read errors. In that case, the cure is just rebooting (power-cycling to reset the board). And if that doesn't work and the board doesn't come back up, then start pulling disks one at a time until things inprove. That just tells you something about consumer-grade hardware (which is often but not always correlated with SATA): It's junk, and has bugs.
There is a reason enterprise-grade computers are more expensive; most of the reason is better quality control, like bug-free firmware.
