For the first time since the famous marque started running two cars in 1979, both will be on the back row of the grid for Sunday's race at the Yas Marina circuit.
Pastor Maldonado lines up 24th on the grid despite setting the 17th fastest time in qualifying, with the Venezuelan dropping down as a result of the 10-place grid penalty he picked up for exceeding the limit of eight engines each driver can use during a season.
Team-mate Barrichello, meanwhile, has been hampered by an oil leak detected during final practice which Williams were unable to cure in time for qualifying. The Brazilian therefore starts 23rd.
A separate engine problem also forced Barrichello out of first practice on Friday. The 39-year-old said: "You have to laugh, not in a ridiculous way, but a positive way.
"I've not driven in first and third practice, and also qualifying. Maybe they (the team) are trying to give me an early holiday, not that I want one.
"It's tough, but at this time you have to smile and support, and hope they solve the issues.
"In the past I would have slammed doors when something like this happened, but it didn't help, so you just have to stay calm and try to fix the problems."
Maldonado also put on a brave face as he said: "We need to come up with a strong strategy to get a positive result.
"We've obviously been thinking about our strategy all weekend because we've known about the grid penalty for a while now.
"We haven't decided 100%, so we'll keep working on it to make sure we give ourselves the best possible chance.
"I will do my best and I know the team will too. We must be confident."