They instead face a huge task just to bat out the day for a draw, with a minimum of 63 overs remaining, though there is a chance rain could come to their rescue.
Their plight could have been worse. Stokes was dropped by Marcus Harris on 16 just before the interval off Pat Cummins.
Harris, fielding at short leg, had two grabs at the ball but could not hold on to the clear chance.
Cameron Green grabbed the biggest breakthrough of the morning with the wicket of Zak Crawley, who had made 77, just as the English opener was looking like he would score a hundred in a flowing performance.
The young giant ripped a yorker under Crawley's bat, striking his boot in front of the wicket and the Englishmen was given out and then lost the review.
Crawley impressed during his 100-ball knock with 13 fours, batting with positive intent and doing more than enough to keep his opening spot with Rory Burns likely to return for the final Test after another failure from Haseeb Hameed.
Green continued to impress with his seam bowling and has now taken nine wickets in the series, second only in the bowling averages at 13.77 to Boland (8.66).
Hameed, dropped by wicketkeeper Alex Carey, went three balls later when he was caught by Carey off Scott Boland for nine.
Hameed's opening spot is in serious doubt for the fifth Test in Hobart in a week's time after a string of low scores.
Dawid Malan was next to go, for four, deceived as he tried to cut a quicker ball from spinner Nathan Lyon that skidded on to his off-stump, leaving the tourists at 74 for two.
The highest winning run chase in the fourth innings at the Sydney Cricket Ground was 288-2 by Australia against South Africa in 2006. England's best is 194 for five way back in 1903.