The solution is to keep a history of backups.

You can store one daily backup, say for the last seven days. Then one backup per week four times per month. This way, if the backup from yesterday had been saved in a bad state, you take the backup from the day before. Or you can take the backup from last week.

To save space you can either use a file system with supports deduplication, use hard links or store only the difference between the backup. Which solution is best, depends on your needs, setup and the software you run.

EDIT: You updated your question and added additional information.

As you already know, you have to separate the data from the backup. A backup is always redundant, if possible even more than one copy. I don't know your NAS solution and their backup software. But I can tell you how I solved this.

I use an old 300MHz system as backup server, which is connected to the file server (that would be your NAS in your configuration). Once per day the backup server switches on and pulls the backup from the file server and writes the data on its own hard drives. As backup software I use rsnapshot. No client computer has access to the backup server in any way. And it is only running for a short time per day.

This is only one possible solution out of many. The key points of a good solution are:

• Keep a history of backups
• A backup is always redundant
• A backup is stored on different hardware (e.g. a second drive, not a second partition on the same drive)
• The client computers must not have access to the backup
• The backup should be as easy as possible, at best fully automatic
• Depending on how often restores are expected, it should not be too big of a burden to restore the data

Is a home NAS reliable to keep your [old but important, private use, no copies on other machines] data on (no other backups)?

Noooo!

Don't even think about doing this, unless you fancy losing your data. Since you refer to the data you intend to store there as "important", I imagine you don't.

RAID will (within limits) help you with recovering from hardware failure, assuming of course that you catch the problem in time and address it promptly. As you quite rightly point out, RAID is not a backup. Also note that all RAID levels are not created equal; RAID 0 provides no redundancy, and can quite easily lower your resilience to hardware failure as all disks in a RAID 0 array must be functional in order for the array to be functional. Lose one, and you lose it all. RAID 5 provides one disk's worth of redundancy, but with today's large disks, the probability of a second failure during the (strenous) resilvering process after a failure is not insignificant.

A "home NAS" is not inherently more reliable than a plain PC built using the same components (and likely is effectively a PC built using very similar components; if you are lucky, with the addition of ECC RAM). It can be made more reliable, but so can your regular PC. (With the exception of perhaps a laptop, for reasons of physical space and hardware compatibility limitations.) For example, my PC has UPS, ECC RAM, and redundant ZFS for storage, all of which contribute to greatly increase storage reliability and all of which are achievable on ordinary hardware. Automated file system level snapshots (ZFS makes this cheap) and automated regular backups provide me with options for disaster recovery.

A single copy of anything, including for example two copies on mirrored disks, is susceptible to being lost to any number of problems. A few examples are (some of these are borrowed from the above linked Server Fault answer):

• File corruption
• Human error (deleting files by mistake)
• Catastrophic damage (someone dumps water onto the server)
• Virus
• Software bugs that wipe out data
• Hardware problems or firmware bugs that wipe out data

If you care about your data, you need at least two distinct copies (including your main working copy); preferably, if you can wing it, three copies (one that you work with, and two for safekeeping). One of these copies should be in a different physical location and ideally one should be on a different type of media than the others. The "distinct copies" part rules out RAIDing the copies; the idea is that a single problem, no matter what kind, must not be able to take out all copies. For a private individual the 3/2/1 scheme (three copies, two backups, one off-site/one on alternate media) is not always practically achievable, but even "just" an external, USB- or eSATA-connected hard disk, hooked to the storage box and regularly updated along with some form of revision history, will go a really long way.