Yes, planes are designed to be able to take off with a lost engine. Usually this will extend the roll a bit because the speeds are different for engine out operations. This isn't the first MD-11 with an engine out take off, 5 years ago a FedEx MD-11 took off with a failure in the left engine[1]. Slightly different case, obviously, but it's certainly something that is accounted for when designing planes.
That being said, depending on how you lose the engine it can really mess up the takeoff; AA191 was lost when an engine detached from the plane on takeoff and took out part of the wing and hydraulic system with it. https://en.wikipedia.org/wiki/American_Airlines_Flight_191
> when an engine detached from the plane on takeoff...
https://cdn.discordapp.com/attachments/757091156717862935/14...
Source: https://reddit.com/r/flying/comments/1ooms7t/ksdf_accident/n...
Yes, the takeoff roll will be longer, the climb will be much more shallow, but it is possible to take off with one out of two engines (and obviously also with two out of three). Of course, after successful takeoff, the plane should turn around and land as soon as possible.
In this case however, with the wing already on fire (the engine is below the wing, so flames coming out of it would be visible behind and under the wing, not in front), I'm afraid that even if they had managed to take off, the fuel tank would have exploded or burned through the wing before they would have had a chance to land. Actually, this looks similar to the 2000 Concorde crash...