I'd also like a car like that but I'd need to sell all I have and/or come into some money to do so. As I'd like to do some VSCC events, it would need to be pre WW2. If this isn't an issue for you, then it widens your scope.
MGs are well catered for in terms of spares and there are always MGs for sale. The MG TD is perhaps the cheapest of the T types, though they have gone up in recent times. It's the first of the T types to have independent front suspension and (I'm pretty sure) rack and pinion steering.
The fifties Morgan, with the TR engine, was quite a machine but Morgans are pricey. Oddly enough, the ones I would like (Pre war series 1, with Coventry Climax or Standard engine) seem to be cheaper if rare.
I reckon the cheapest car of this type, that you'll find, is some sort of sports Austin 7. The Nippy is cheaper than the Ulster and has the advantage (You mentioned your health) of having doors; in the Ulster you have to climb over the side, albeit a lowered side. I think they are pretty well catered for in terms of spares and tuning. I saw some engine blocks for sale at a VSCC event a few years back and they looked almost small enough to fit in my pocket

Despite this, they looked good fun on the track.
At about the same time, a friend of my father's died. He was 69 years old and had had about 70 cars during his life (He knew he was dying and had bought a nice MG TD as a final fling), from Austin 7 sports (Not sure if it was an Ulster, Nippy or Speedy) to V12 E Types and Aston Martins. They were always "Classic" when he owned them and he liked to try something new. He always said that the Austin 7 was a lot of fun; just as much fun as the bigger faster cars, just at a lower speed (Though it didn't always feel slow).
There are others like Riley, HRG and Singer Le Mans but these are going to cost as much and more than the MGs. The Automobile Magazine caters for older cars and here are some interesting links that I use to “Window shop”.
http://www.barrywalker.com/
http://www.terrybone.co.uk/car.php
http://www.mgsparesandrestorations.com/home.cfm
http://www.prewarcar.com/ This is connected to "The Automobile" Magazine.
http://oldcarservices.co.uk/index.php?page=bodywork
http://www.pwa7c.co.uk/
http://www.vscc.co.uk/vsccweb/
Good luck with your search

Rules exist for the obedience of fools and the guidance of wise men.
MG Midget 1500, MGB GT V8, Morris Minor Traveller 1275, MG Midget 1275 & too many bicycles.