The doctor says a couple of glasses of red wine several times a week would be good for me. Should I have told him that my wine glasses each hold 1/2 bottle? 
Not drinking the whole bottle? Hell, I toss the cork the second it clears the neck. It’s done its job…no need for it anymore. Sorta like the lid on a pint of Ben & Jerry’s.
Even better, buy a screw-cap bottle and use it for this purpose (don’t forget to rinse it out well between different types of wine). There are quite drinkable wines available in screw-cap bottles these days- Bonny Doon Vineyards and Tin Roof make some nice ones in California, and there are some good ones from Australia and New Zealand.
Whether you go with the half bottle, screw-cap bottle, or just re-corking the original bottle, do store the wine in the fridge. Don’t store it with the bottle tipped over on its side if it’s a re-corked bottle.
I’m now going to reveal myself as a wine barbarian. Watch in awe. (This isn’t going to help the OP, but it’s definately MPSIMS.)
My SO and I do a little bit of wine making from kits. Knowing that we often won’t finish a full bottle, we’ve started bottling in long-neck beer bottles. They nicely hold 2-3 glasses of wine. If we’re feeling especially lazy, it can be drunk straight from the bottle. Anyone else want to swill the house red from a long-neck?
My BIL, who is a winemaker, says not to store reds in the fridge. It might be OK for Whites, but not Reds.
The screw tops are great, but once you’ve introduced air into the wine bottle it will have an effect on the wine. It’s better to eliminate as much air as possible. Screw top half bottles would be best.
One largeish glass a day = four days to empty a bottle, which means that the last glass on day four might be starting to go a bit offish, but is still drinkable. I only open a bottle if I know I’m going to be home for the next four days to drink it, or if I don’t mind throwing some of it away.
On the other hand, a bottle is exactly the right size for me and a friend to have two glasses each on Friday night, get nice and tipsy, and watch movies, which is what I often do with a bottle of wine.
Amen, I find putting wine away to be a sign of weakness of character. Unless of course it was weakness of character that opened a third (or fourth) bottle of wine between Mrs Prefect and me 
The OP isn’t talking about storing a wine long term in the fridge. They’re talking about preserving an open bottle. Reds will be fine in the fridge for this. But like everyone said, once the bottle is open, no wine is going to last longer than 3 to 5 days.
This thread is already filled with excellent advice on preserving an open bottle of wine, but what the hell.
This device is top notch, but not everyone is willing to spend that kind of money.
My personal choice would be a can of Private Preserve and a stainless steel cork - cheap, extremely efficient, and super easy.
A Vacu Vin is another cheap and efficient method to preserve leftover wine.
Again, with any of these methods I’d store the wine (red or white) in the fridge, and expect to drink it within 5 days max, after that you might be able to cook with it, but if it’s not used after a week or so, just toss it.
If you reeeaally can’t drink a bottle of wine in a couple of days, you can use the excess to cook with.