Imagine an A320 on the runway, starting its take off roll. The PM calls out "$V_1$". Just a second after $V_1$ is called out, and before $V_R$, you encounter a double-engine failure (an A320 has only 2 engines). Now, you've got no engines working. You also have less than the runway length required for you to decelerate on the runway. What should you do next?
The Airbus A320 FCTM (Flight Crew Training Manual) does not describe any procedure for rejecting a takeoff after reaching V1:
The decision to reject the takeoff is the responsibility of the Captain and must be made prior to V1 speed
Nothing further is mentioned about rejecting at a later time. However, the FCTM and also the QRH (Quick Reference Handbook) never even consider a dual engine failure during takeoff. Only a dual engine flameout at high altitude is considered. Not even a dual engine failure at low altitudes is mentioned, although my copy of the FCTM is a bit older, so it does not include the changes made after AWE1549 (the miracle on the Hudson). The final accident report says
US Airways’ dual-engine failure training, which was provided during initial training in a full-flight simulator session, was consistent with the training provided by Airbus. The dual-engine failure scenario was presented at 25,000 feet, included two engine restart attempts, and was considered complete after the restart of one engine [...]
No dual-engine failure training scenarios were presented at or near traffic pattern altitudes [...]
I assume they made some changes after this accident.
The Boeing 737 NG FCTM actually mentions rejecting above V1:
Rejecting the takeoff after V1 is not recommended unless the captain judges the airplane incapable of flight. Even if excess runway remains after V1, there is no assurance that the brakes have the capacity to stop the airplane before the end of the runway.
Rejecting after V1 is only allowed if the airplane is incapable of flight. This is certainly true after a dual engine failure before VR because you cannot climb safely below V2 (and you cannot maintain V2 anyway). Even the attempt to rotate the plane before VR would likely result in a tailstrike.
The Boeing 737 NG FCTM further shows the effect of a late reject decision:
As you can see in this example case, where the rejected takeoff (RTO) was started 2 seconds after V1, the airplane would be moving at 75 kt at the point where it would have come to a complete stop if RTO had started at V1. The airplane then comes to a complete stop 210 m later.
However, this does not necessarily imply that the airplane leaves the runway. That is only the case if V1 is limited by runway length. If V1 is instead limited by Vmbe (maximum brake energy), there would be more runway available, but the brakes might fail before coming to a complete stop. The best case is if V1 is limited by VR, which might give you both enough runway and enough stopping power to come to a complete stop (your scenario describes the failure between V1 and VR, so this cannot be).
In any case, the best decision if the airplane is incapable of flight is to attempt an RTO. It is better to crash at low speed than to crash at high speed after trying to get airborne. The runway might even be equipped with an RSA or EMAS, which reduces the risk of severe damage and helps to stop sooner.
One-engine failure (OEI) and all-engines failure are not approached the same way.
Today an engine failure during takeoff is an event with a very small likelihood. Engines are tested and proved to be resilient, bird strikes and FOD ingestion are made infrequent by prevention. Still this can happen.
On the other hand, a double engine failure during takeoff has a likelihood which is insignificant.
The different likelihood associated with these two cases are not mitigated the same way:
The event with the highest likelihood, the single engine failure, is reduced by the takeoff protocol involving V1 speed and allowing the crew either to stop safely before V1 or to takeoff safely after V1. The price to pay for this safety is it requires a longer runway. This constraint has been accepted by airlines and airport managers.
The double engine failure preventing to take off, has no economically viable solution. It was decided runways all over the world wouldn't be extended to allow a normal and safe stop at a speed higher than V1. So the aircraft must be stopped using all braking means available. And if this is not possible, the aircraft will leave the runway. Airports are designed to not make this accident worse than it is, e.g. obstacles like lights, signs or poles are required to be frangible at ground level..
However some last resort feature can be put in place at the end of the runway to stop the aircraft with exceptional means. This can be a safety net for military, or an EMAS for civil aircraft:
EMAS are useful both for takeoff and landing, to prevent runway excursions. So they tend to be more acceptable economically. There are several questions on the site about EMAS. However EMAS and similar arresting system are not full guaranties for a safe stop, the aircraft has still to roll in straight line to reach them.
For events that are assessed and found to be very unlikely, there is no specific solution except prevention, crew best judgment and skills to manage the unexpected. This is not different from other domains.
What would the crew do? They would brake using all possible means and use their skills either to maintain the aircraft on the runway, or on a taxiway, or anything they can do.
@Bianfable has detailed some of the guidance available in his answer.
In this scenario, many elements have been anticipated, random examples:
Thrust reversers would be inoperative, but aircraft demonstration for certification requires to brake without reversers.
Brakes would likely be excessively solicited and possibly a fire can start. Aircraft certification requires this fire to be contained to the wheel for 5 minutes, the time for firefighters to access a crashed aircraft
The tires mustn't explode but deflate.
Question: What should you do next?
I am not aware of any published procedure that has been written in anticipation of the unlikely dual-engine failure scenario in the OPs question. But, in my opinion, I would think the following actions would be reasonable:
Thust levers closed, auto throttles disengaged, apply max braking (or if auto-brake RTO is set ensure it's working correctly), ensure all spoilers/speedbrakes are fully deployed, apply max reverse thrust (for any potential benefit of residual reverse thrust during engine spool-down), tell the cabin to brace for a crash and advise ATC so emergency response can be initiated.
Do not attempt to fly a plane incapable of flight!
What goes up must come down. The "coming down" will be Very Bad, and occur probably at destructive vertical speeds and definitely at a site not engineered for that purpose.
As my martial arts teacher says about falling... you are already on the ground. So is a rolling airplane. While you may not be able to stop in runway length, your collision energy will be much lower by the time you exhaust the remaining runway, because you have used that remaining runway for max braking.
And most runways have been engineered for runaway overruns, with crushable things like "breakaway" lights and antennas (a great deal of experience crossed over from highway design) inside the fence... and easements outside the fence so that hard things are not built along the extended runway center line.
If you have reached 88 miles per hour, immediately activate the flux capacitor and quickly select a year.
If both engines have failed there is no choice to make. You aren't going flying. Abort immediately and you will eventually stop... somewhere. Brake at maximum capability, ensure spoilers deployed automatically when you pulled the power to idle.
The thrust reversers won't have any thrust to reverse. It could be argued/debated that deploying them could create some minimal drag/cancel any residual thrust while the engines spool down.