Right now the state of the art with DVR technology is that they're best for recording live TV programs for playback fairly soon after the broadcast. It looks like you need something for archival purposes.
I too have a collection of VHS (and U-matic, Betacam etc.) tapes that I want to preserve. What I use is a Grass Valley Canopus ADVC110 to capture the NTSC signal from the various tape players to my computer, making a D-1 video file that's then compressed for storage, in my case on a RAID array.
I should say that I'm a long time TV broadcasting professional, so my methods are probably more involved than what you need. I've grown tired of keeping track of hundreds of optical discs, so I just keep them on a network file/video server that can stream the files' content to my TV. This is leading edge stuff, and may be too complex/costly for you.
If you're most comfortable with keeping your programs on optical discs, then you can use a computer (I'm guessing that you have one, since we're talking on the Internet) to capture the VHS tapes, and then make Video DVDs on your computer. Inexpensive capture devices from Pinnacle and StarTech can be found at the local big box store, though I would look for Hauppauge devices; I've had good results with that brand. DVD writers for computers are dirt cheap. Chances are that your computer already has one, but if not you can get one for next to nothing. Most of the capture devices come with software that will do an adequate job, and there is freeware on the Internet to choose from as well.
All told you can spend from $50 to $200 for all you need to do it with a computer. Maybe a bit more if you don't already have a good VHS player.
By buying the various components rather than having them all on one chassis, if one piece breaks down, you don't have to throw out the whole thing.
If you are growing tired of having lots of DVD discs, you can usually save the video as files, and buy an inexpensive HDD to store the files. I use RAID arrays so if any one HDD fails, I don't lose anything, but you can add reasonable protection by using 2 or more drives to store your video files.
What sort of computer do you have? Desktop? Laptop? PC or Mac? I could suggest a few setups if you like.
Also, I'd like said DVR to have the capability to record pay-per-view movies that I rent from my cable company (Charter Cable) as is my right, IIRC, from my Mass Communication classes in college.
IIRC, according to FCC regulations, if you rent a movie from your cable company you're entitled to record it and watch it as many times as you like. But,(of course), the DVR that game with my cable package DOES NOT record movies you rent.
So my question is: is there a consumer DVR capable of recording rented movies, and is there a way I can transfer my old VHS tapes to it's hard drive?
The terms of your contract with Charter are very long-winded and beyond the scope of this discussion. To make a long story short, any DVR that works with Charter has a "CCI flag" setting that limits your ability to copy the DRM content, be it PPV or from a premium channel. That has nothing to do with dubbing VHS recordings to digital media.
Now if any of these VHS tapes are movies with Macrovision copy protection, you will likely encounter problems trying to dub them using readily available equipment. In that case my advice is to buy the DVD (or Blu-ray) edition of the movie, and accept that you're not going to get a rebate because you have a VHS copy. Trying to defeat copy protection can open you up to lawsuits and possibly criminal charges, so it's better to stay away from piracy no matter what a college teacher said.
I've been using TiVo DVRs since they became available, and have a substantial investment in them. I also have software that allows me to take non-CCI video as files, and store them in my library. The last time I was on Charter, they made it very difficult to use my TiVo on their network. They would have preferred that I had rented their equipment, but since FCC rules say they must accommodate me nominally, they had to comply. But it wasn't easy.
Neither a TiVo nor a Charter supplied DVR will allow you to dub from VHS to them. Older non-HD TiVos had the ability to dub from VHS, but it was limited. I chose to use my computer because I found dubbing to a TiVo (years ago) was too cumbersome. You may find some no-name DVR that can dub old NTSC recordings, but I doubt that it will have a CableCard slot, so you'll never get it to work with Charter.