If you're a small non-profit...especially the small part...there's really not a reason to move to the cloud. What is the current deficiency in your backup system you have now?
You said you're doing backups to tape yourselves; what are you paying your IT outsourcer to do that you're paying an arm and a leg for? If the backup solution is properly implemented it should be largely automated. You should be paying your IT help people to maintain and troubleshoot issues, not maintain your backups. If you are paying them to come in and do backups, then A) you aren't using the right solution or B) they're working on other things and you're kind of alluding to two issues, and I don't know how the backup change will solve your large IT support bills.
For file sharing and backup, a good non profit solution is to invest in a system to run OpenFiler or FreeNAS for file storage, and another system to rotate storage to tape off the file shares since you already said you had a tape backup system in place. Openfiler and FreeNAS both support software RAID and you wouldn't need Windows licenses.
If you can get a deal on two decent systems (which aren't all that much for a five-person outfit) you can get two systems and run an automated unison or rdiff-backup to synchronize data on two Linux systems.
You may need a tech person to come in and get you set up initially, but for the most part as long as the hardware doesn't fail much of what you describe should be really easy to automate for the most part by a qualified tech person so you don't need to pay as much for periodic visits, if that's what is costing so much to your organization.
Also some prevention can go a long way...i.e., decent hardware for the servers (backup and file storage) so it doesn't crap out on you, some form of RAID that isn't in the motherboard (either software based on Linux or hardware based like 3ware for linux or Windows), and UPS's to protect your hardware will help you in the long run.
I know I'm suggesting spending money in order to get something up and working well, but without knowing what hardware you have in place and what system you have in place right now it's hard to recommend anything. I also don't know what your arrangement is with your IT people. For a five person group you could probably get away with spending on a couple systems off-the-shelf to act as a file server and backup server like I'm proposing. Or you may be able to settle for a single computer running Linux with a couple external firewire disks on which you can save files with diff-backup periodically so your external drives are your backups and your system is a file server.
It also largely depends on what kind of data you're storing...a small database?
I'd almost recommend a seven hundred dollar system or so with Linux and a couple external hard disks on which you save data as a backup to one and copy it to the other external disk in case of failure...that should have you more than covered for simple needs.