It's also an idea I kind of cringe at hearing because you have people who aren't probably "Tech people" handling hard drives and traveling in their cars with them. You don't mention the amount of data you're backing up, or what business you're in, or even how you're making the backups (copying files? differential with a particular program?...) Is it encrypted? People who aren't "tech" people aren't often known for taking care not to cook equipment in the back seat or making sure everything is locked up all the time.
You also don't mention your options. You mention your limitations, but not options...for example, is there a small office space or company owner with a fast business connection at home? Data could be copied over VPN to a server offsite, which would automate things to a huge degree and eliminate the "whoops" factor. Depends on your data needs, though. I'd mention a company owner because that would also eliminate the awkward firing an employee with the backups thing.
The switching of 5 drives is definitely overkill. You really only need one offsite storage; one person with the duty simplifies the management and tracking of resources and responsibility.
The lack of money part is a common complaint but really, it comes down to how much it's worth to have your data recoverable. How much money does the business lose if the server dies? How much will it cost if the data is lost completely? If the pursestrings can't be loosened or money found to insure recovery from those scenarios, then there's another problem to deal with. No business wants to spend more than it must.
My personal guidelines are these:
The simpler, the better.
The fewer people involved, the better. It's more reliable to have one person in charge of having this done than 5 people running around with data.
Encrypt it. Especially if it's leaving the site.
Automate it. If you can use a VPN solution to link to an offsite storage server then you can delegate a person to periodically just check the logs to make sure the backups were successful.
If you're going to continue the single hard disk option, narrow it down to one person taking it offsite. Keep one drive on site. Swap which drive is taken offsite. Periodically check the drives for disk errors.
Have you taken into account problems with the server? Again, depends on how much you rely on your server and how much downtime you can stand, but if the power supply dies or controller/network card fails, how long will it take you to get working again? Can you get a second system to set up as a mirror of the first, or a supply of parts should the first system fail?
Make sure if you're transporting drives that you have a good carrying container for it (preferably a cooler with insulation/foam in it to prevent wild temp changes and bangs from damaging the disk platters). Maybe it will also be slightly less appealing a theft target, as people might be more inclined to steal a briefcase than a lunch.
Weigh your situation against how much it costs to lose the data, how much you want to invest in managing it (remember that the more complicated it is, the greater the chance something will go wrong, or the greater the chance that other people won't follow through or find shortcuts that will lead to problems), and what scenarios are likely in your situation to protect against (dead drive, corrupted data, theft, flooding...) and then come up with a DOCUMENTED procedure to follow. DOCUMENT your backups, document how you're doing them, who to contact, who's in charge of what, even parts on the server. Imagine (especially in a small business) if a new "IT person" were hired and on their second day your server died, and no one else was around. You should have a binder that they can flip through with all the information needed to get the business up and running again.