Tape doesn’t really sound suitable for your project - any decent sized tape drive is going to be outside your budget. I’ve got a Seagate (now Certance) DDS-4 drive, which costs around $700. You need a SCSI card. Tapes take 20GB.
However, the media costs are likely to be higher than DVD-R, the tape is pretty vulnerable (I have had one bad batch of tapes snap when ejected and had to run extensive qualification tests to find a tape brand that works reliably - hint - Seagate’s own media certainly was rebranded FujiFilm). The media is highly vulnerable to damage through bad storage conditions - there’s 150m of 4mm wide tape in a DDS-4 cartridge, and the drive uses helical scan, like your VHS machine. That makes the tape itself more vulnerable to damage than linear formats (like audio cassettes and some of the higher end tape formats like DLT and Ultrium).
The only way you can access tape drives is through backup software. Drive letter access just doesn’t work reliably - the seek time of tape is just too slow (can be a minute to seek from one end of the tape to the other).
Anything bigger than 20GB per tape is going to be seriously expensive. There’s now a 72GB DDS format (which, in truth, is 36GB - the headline capacity assumes 2:1 compression is possible, which is rarely true for any data and definitely not true for already compressed data like anything in one of the MPEG formats). Though the drives aren’t much more expensive than DDS-4 units, and can operate on DDS-4 media as well, the “72GB” media is still very expensive and the format really unproven.
Other tape formats have significantly more expensive media costs than DDS. There are 200GB headline formats, but you’ll pay somewhere up over US$3000 for the drive, and several hundred dollars per tape, whilst only getting around 100GB per tape.
For the right application (backup) you just can’t beat tape. You can keep overwriting the tapes - though the cartridge life of DDS can be lower than you think because of the damage imparted to the tape itself from the helical scan mechanism.
That said, the future of DDS could be bleak once we have an affordable rewritable 20GB+ optical format. The more ‘enterprise’ tape formats will survive - but much as I’d love a big backup device like an Ultrium 2 drive, neither the drive nor the media are affordable for SOHO use!
The best archival policy is to write at least two copies of the data to two different types of quality DVD media (possibly one brand of DVD-R and one brand of DVD+R - look at the reviews and forums for recommendations of what is a quality product on your drive).
Before putting each disc away you should attempt to scan it for quality, using DVDINFOPro or similar. If you do a read speed test using something like Nero CD-DVD Speed, that will give you a better feel of the quality of the write (ideally check the disc in a drive other than the one you wrote it in).
You should then revisit the discs every six to twelve months, recheck them and ideally write a new copy to a fresh batch of DVD media - thereby guarding yourself against degradation of the media.