TL;DR: 7 Levels are enough

1.) A Monk 10 / Fighter 2 can do both.

Let's use the the following approach. Take Mobile as a Human Variant Starting Feat, take two Figher Levels to get access to Action Surge, then take Monk Levels for Unarmored Movement. You can then use Action Surge and Step of the Wind to Dash three times in one round, which gives a total movement of

$$ (6+2+k)\cdot4 $$

squares where \$6\$ is the Human base speed, \$2\$ comes from the Mobile feat and \$k\$ from Unarmored Movement (the amount of which we leave variable in order to determine the least amount needed to break the world record). A round is \$6\$ seconds long and a square corresponds to \$5\$ feet which is \$1.524\$ metres. Hence, even if you have the maximum of \$k=6\$, this will only take you

$$ (6+2+6)\cdot4\cdot1.524\approx85.344 $$

metres in one round. Thus, we need another round to cover the remaining

$$ 100-((6+2+k)\cdot4)\cdot1.524 $$

metres. In this round however, Action Surge will no longer be available, leaving us with

$$ (6+2+k)\cdot3 $$

squares of movement, which is a speed of

$$\frac{((6+2+k)\cdot3)\cdot1.524}{6}$$

metres per second. Putting things together, this strategy lets you run \$100\$ metres in

$$t(k)=6+\frac{100-((6+2+k)\cdot4)\cdot1.524}{\frac{((6+2+k)\cdot3)\cdot1.524}{6}}$$

seconds. Plugging in the possible values of \$k\$ yields

$$ t(2)\approx 11.123, \quad t(3)\approx 9.93, \quad t(4)\approx 8.936, \quad t(5)\approx 8.095, \quad t(6)\approx 7.374.$$

Hence, a Monk 9 / Figher 2 is still quite a bit away from beating the world record, while gaining another Monk Level already makes her superhuman at a total Character Level of 12.

2.) And so can a Monk 2 / Fighter 2 / Barbarian 3.

What follows is a more detailed and precise elaboration of the idea suggested by user48255. Again, take Mobile as a Human Variant Starting Feat, take two Figher Levels to get access to Action Surge, take three Barbarian Levels to get access to the Elk Totem Spirit, then take Monk Levels for Unarmored Movement. While raging, you can then use Action Surge and Step of the Wind to Dash three times in one round, which gives a total movement of

$$ (6+2+3+k)\cdot4 $$

where \$6\$ is the Human base speed, \$2\$ comes from the Mobile feat, \$3\$ is the Elk bonus and \$k\$ the Unarmored Movement. Since we can only take 20 Levels, \$k\$ is bounded by \$5\$ in this case and since

$$ (6+2+3+5)\cdot4\cdot1.524\approx97.536 $$

isn't quite enough, we will again need another round. Again, we won't be able to use Action Surge in that second round, leaving us with

$$ (6+2+3+k)\cdot3 $$

squares of movement. In analogy to the above, this strategy lets you run \$100m\$ in

$$t(k)=6+\frac{100-((6+2+3+k)\cdot4)\cdot1.524}{\frac{((6+2+3+k)\cdot3)\cdot1.524}{6}}$$

seconds. Plugging in the possible values of \$k\$ yields

$$ t(2)\approx 8.095, \quad t(3)\approx 7.374, \quad t(4)\approx 6.749, \quad t(5)\approx 6.202.$$

Hence, a Monk 2 / Figher 2 / Barbarian 3 can become superhuman already at a total Character Level of 7. You do have to take care of sustaining your rage, though. Maybe put a d4 on the floor and step on it. ;-)