I have three options I'll present for you. One matches your headline, one is cheap, and one is really, really solid.
The option that most matches your headline would be to use a WIFI NAS at the next door neighbor's house. Use any of the many good backup software packages. More on what a "good" backup system is in a moment.
Something I used to do was have two external drives. On Mondays, I'd switch out the drive in the house for the one in the car, which would go to work with me. The drawback to that is it's not fully automatic, so sometimes I'd forget or be in a rush. That leads us to the attributes of a good backup system:
Backups must be fully automatic, otherwise you'll stop doing them regularly.
Backups should be rotated. A midnight backup is useless if you are hacked at 11:55 PM, or discover a problem 2 days later. You must have access to older backups.
Backups must be offsite. Fires and burglars will take your backup if it is on site.
Backups must be accessible. As you said, spending two weeks downloading your data isn't acceptable.
Backups must be tested. Our experience with web servers indicates that approximately 60% of backups provided by hosting providers don't actually work when you try to restore them
To meet all of the above requirements, we use an enterprise grade system. It may be overkill for your needs, but then again the $8 / month version may be just what you want. It provides several offsite backups from different points in time and they are BOOTABLE. You can pull down a file or two, run a program or service remotely, or restore a full system.
3-4 Mbps to transfer 1TB is no good, as you said, but you actually have 200 Mbps available if you use the system we use. If you need the entire 1 TB, not just a small part of it, the whole 1TB bootable drive will be delivered to your front door within 12 hours. You may know the old saying "never underestimate the bandwidth of a station wagon full if tapes.". With a 1TB drive, the bandwidth of FedEX is over 200 Mbps.
What we use is called Clonebox. It's designed more for business, but it may either work for you, or give you some ideas.