You can afford a decent property in basically all parts of the country. The healthcare services are affordable and in a good quality. With an average British, German or Dutch pension, you can afford a clearly better quality of living than in your home country, whilst enjoying the mild climate as well. In Hungary all conditions are given for a peaceful time in the retirement age.
The countryside is scenic no matter whether you have a holiday house at the hills of Balaton, near the Austrian border, the mountains around Budapest or the Great Hungarian Plains.
Due to the low population density, there is a lot of space available for your dog- and you don't even have to pay taxes for this! Also, there is an ancient tradition of respecting horses (due to the legend, the forefathers of the Hungarians arrived on horseback and they even traded a horse for the ownership of the land), you can find everywhere riding schools and stables.
As the prices are very low comparing to Western Europe, it is very likely that they will be catching up in the future. The country is a great place for investment!