"doomed quest" is correct and usable in written English.
You can use it to refer to a seemingly impossible mission, journey, or search. For example, "The explorer's doomed quest to find the hidden city ended in failure."
Like many a Pynchon protagonist before him, Sportello is on a doomed quest.
Timothy Mackabee's stylishly sombre set amplifies the disparity between Victorian mores and Merrick's doomed quest for normality.
There are no shining heroes here, but the quietly frantic striving for tenderness comes off as a noble, doomed quest.
"Freud said: 'The pursuit of happiness is doomed.' It's a doomed quest because it's fuelled by infantile fantasies," added Perry.
The Impressionist, by a Briton, Hari Kunzru, is a picaresque Bildungsroman, whose shape-shifting hero traverses continents in a seemingly doomed quest for home.
When Irina finally finds Aleksandr, the year is 2006 and he has given up chess and embarked, à la Garry Kasparov, on his own doomed quest — a presidential campaign against Vladimir Putin.
It is in telling the profoundly human stories of these often doomed quests that the small but richly evocative Polar Museum excels.
McCandless was a follower of Thoreau, but his story is best understood as part of the tradition in which doomed quests for a hidden God end in images of uncanny whiteness.
In books like "In Search of Love and Beauty," "Three Continents," "Shards of Memory" and "East Into Upper East: Plain Tales From New York and New Delhi," Jhabvala writes about the post-Nazi generations of European refugees and immigrants in America: a gloomy subject she often enlivens by sending some of her characters to India on farcically doomed quests for spiritual wisdom.
That they are doomed, their quest for a repeat squashed?
Michael Lawler and his girlfriend Barbara Burdick were sailing their 47-foot yacht Traveler from Salalah, Oman — where the doomed S/V Quest was heading — into the Red Sea when three pirates aboard a small fishing boat approached them, armed with an AK-47 assault rifle.
