Just FYI, the home used in "Downton Abbey" is called Highclere Castle, and is the home of the Earl and Countess of Carnarvon. Here's their website

Some do. The Duke of Marlborough still lives at Blenheim Palace, which is about as splendid as you get. But many in the British peerage just couldn't afford the expense of maintaining such expensive homes during the economic changes in the 20th Century, with a large number of them signed over to the care of the National Trust to preserve.

Although some British aristocrats still live on splendid estates, fewer and fewer do. It has become difficult to make the economics of large landed estates work.

Two reasons: Changes in the economics of agriculture and trade and changes in taxation.

The "Corn laws" which protected cereal crops from foreign competition were repealed in 1846. This did not have an immediate impact on British Agriculture, but by the 1870s, bountiful American agriculture in the midwest, combined with railways and steamships providing cheap transportation to Britain, meant that grain prices started to fall.

This hurt the large estates which relied on growing grain. (Smaller farmers, growing fruit, vegetables, dairy cattle, were not hurt badly, as these products could not keep well enough to be transported over long distances.)

The Income tax (which had been introduced to fund the Napoleonic Wars and then eliminated) was re-introduced in 1842. Over time the rate of this tax increased, and this hurt the large estates.

In 1853, Death Duties were extended to cover landed estates. Originally only on the annual value of the estate, not the capital value, but in 1894 the Death Duty on land was changed to a tax on the capital value.

The combination of lower returns from arable agriculture, income taxes, and death duties on land, combined to place continuous and increasing pressure on the viability of large estates through the last half of the 19th and the 20th centuries.

That is why so many of them struggled, were broken up, or disappeared.

Sources:

http://books.google.com/books?id=azMJAQAAIAAJ&pg=RA7-PA33&lpg=RA7-PA33&dq=History+of+taxes+on+British+landed+estates&source=bl&ots=uihuxPwJwQ&sig=eLe2qHcEfQYBaB38bRoKrbuPix8&hl=en&sa=X&ei=-zQZU5noIqGSyAGS44GoAg&ved=0CEAQ6AEwBDgK#v=onepage&q=History%20of%20taxes%20on%20British%20landed%20estates&f=false http://webarchive.nationalarchives.gov.uk/+/http://www.hmrc.gov.uk/history/taxhis2.htm http://www.timeshighereducation.co.uk/books/which-straw-broke-the-back-of-british-agriculture/171618.article