Many of your questions are starting to veer off the primary topic of the original question and more into the territory of hiring a consultant to answer your questions, as it's getting very specific to your situation. But here are some points to keep in mind.
Your basic question breaks into, "We have a storage appliance that's going flaky. What can we do to have network storage that won't have this issue?"
Unfortunately whenever you have a single connection to the network from any appliance/server, you risk a bad network card taking it out. For a small business redundancy for network connections is probably overkill.
What you need to do is have in place a system to keep from losing your data, and get your data back online in the event of catastrophic hardware failure. With appliances, unless you spent a lot of money on an appliance with hardware contracts or commodity parts that can be replaced with a trip to a local compu-mart or overnight Amazon/NewEgg orders, appliances are generally not all that redundancy-friendly or service-friendly.
As has been suggested, you can take a regular PC and install a turnkey appliance-like program like OpenFiler to handle your storage needs. You need at least two hard disks of the same size in the computer, maybe three drives (2 for storage, one of a small size) if you want it self-contained. That way you install OpenFiler to drive one and turn drives two and three into a RAID volume (mirror).
OpenFiler (or any self-contained NAS bootable program) is an operating system with a series of scripts and front-end programs to handle the server functions. You supply the hardware, that's it.
RAID creates redundancy, but NOT backup. RAID will protect from disk failure. One drive dies, you continue to get your data, but you NEED to monitor it (or set it up to warn you) if a drive fails. Both disks die or the controller dies, you lose access (or corrupt your data). Replace the bad disk and the software should be able to re-sync everything.
That's out of the way, you need to next consider BACKUP software. You need to keep data stored on a second drive on a separate machine (or attached disks), or a tape drive, or DVD's, or whatever works for you. Why? What if your data is corrupted? What if you have infected software on the storage device? What if a user deletes something that was really important to the business? Backups let you restore data from your last snapshot or further back; you discover that spreadsheet is gone, but you can get it back from a week ago instead of rebuilding the whole thing. Also backups can be designed in such a way that you take your data offsite. If you have a fire or flood, at least your data is safe. (consider encrypting backups to guard against theft or loss; you don't mention your business, but you may end up in legal trouble if you have, for some reason, credit information, information for identity theft, or medical information, payroll...)
Is it expensive? Depends on your definition of expensive. What is the cost of your business and time if you lost your server altogether? Suddenly spending two grand on new hardware for the server isn't so expensive measured against losing your payroll and customer information for a few years.
Last thing to consider...RAID on the cheap can be done. But it's most effective when someone knows what they're doing. You need a consultant or someone in your organization to take this seriously. Why? Because there will come a time when this set-it-and-forget-it arrangement will fail, and if the drives aren't labelled, how will you know which one failed? Replace the wrong one and you'll find yourself needing that backup because you destroyed the volume! One of the best reasons to use hardware RAID, in my opinion, is because most of them have a way of telling you which drive failed. 3Ware cards will tell you the drive on port 2 failed, you open the machine, look at the cable in port 2, follow it to the dead drive. Some of them even have the ability to hot-swap drives and blink lights on drive cages (or the card itself for monitoring) so you can follow where the failure is. The drawback is that open-source turnkey NAS products may or may not have the ability to support hardware volumes integrated in them, so you might instead end up creating a Linux server of choice and sharing files out. That takes more research and work on your part, but in the end is another set-up-and-forget system once configured.
There is no easy way to do this on the cheap if you're not already familiar with the concepts. It's confusing, admittedly, and the worst part is that it's easy to think you have a good solution when it's a ticking time bomb (oh, just set up a share on your coworker's drive and share files there. Map everyone to Bob's workstation as drive F: and you're good to go!). Things will look like they work fine right up until something goes really wrong.
Before balking at cost, look at the alternative of losing that data. Once you know how much that data is worth to you, you should know how much you're willing to spend to keep it safe.