How is minimum runway length related to V1?
Takeoff Field Length
The “FAR” Takeoff Field Length determined from the FAA Approved Airplane Flight Manual (AFM) considers the most limiting of each of the following three criteria:
1) All-Engine Go Distance:
115% of the actual distance required to accelerate, liftoff and reach a point 35 feet above the runway with all engines operating:

2) Engine-Out Accelerate-Go Distance:
The distance required to accelerate with all engines operating, have one engine fail at $V_{EF}$ at least one second before $V_1$, continue the takeoff, liftoff and reach a point 35 feet above the runway surface at $V_2$ speed:

3) Accelerate-Stop Distance:
The distance required to accelerate with all engines operating, have an engine failure or some other event at $V_{EVENT}$ at least one second before $V_1$, recognize the event, reconfigure for stopping and bring the airplane to a stop using maximum wheel braking with the speed brakes extended.


$V_1$ Speed
$V_1$ is the speed chosen by the manufacturer as &quot;the pilot’s initiation of the first action to stop the airplane during the accelerate-stop tests.&quot;  Note that this is particularly important in that it is not a takeoff decision speed as some pilots like to think of it.  The decision must have been made prior to $V_1$ so that there is time to react by $V_1$.
This speed is chosen by the manufacturer and must be between the engine failure speed1 and rotation speed.  Within this range, they could choose any speed, but typically they choose the speed which gives a balanced field length (see next section).
There are also other factors to consider such as the maximum brake energy limit ($V_{MBE}$) which will prevent $V_1$ from being too high.

Balanced Field Length
If we takeoff from a fixed length runway (like, say one that matches the Takeoff Field Length from above), changing the $V_1$ speed will affect the weight that we can successfully continue the takeoff or abort within the available runway.
At a particular weight, a lower $V_1$ will enable us to stop sooner if we abort but will take more distance to continue the takeoff.  A higher $V_1$ will require more distance to stop but will take less distance to continue.  If we graph these values, there will be a point where the $V_1$ speeds for the abort/continue lines meet, and this is known as the balanced field length.  It is the value typically given in the AFM:

This is nice because it gives the pilot one important number to remember instead of different stop/continue $V_1$'s.
If they were to choose an extremely low $V_1$ as in your question, then it would require considerably more runway in order to continue the takeoff than if it were higher, and it would not be balanced.  Some AFM's do include separate charts for accelerate-go and accelerate-stop though so you could figure out exactly how much runway that you would need if the engine failed at a lower airspeed.

Rotation Speed
$V_R$ is determined independently from $V_1$, and if it is ever less than $V_1$ then it must be raised to be equal to $V_1$.

References
The regulatory information for takeoff performance on transport category aircraft is listed in 14 CFR Part 25, Subpart B, Performance, 25.101 - 25.113.
Of particular interest for this question would be 14 CFR 25.107 - Takeoff speeds and  14 CFR 25.113 - Takeoff distance and takeoff run
An easier to read document is the FAA's Pilot Guide to Takeoff Safety which covers this and a lot more.  Much of my answer was taken directly from this document, and I would highly recommend anyone that is flying a multi-engine aircraft to read the entire thing.

1 $V_{EF}$ itself must be above the minimum controllable airspeed on the ground plus acceleration while the pilot reacts to the engine failure.