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 need to break the world record). A round is 6 seconds long and a square corresponds to 5ft which is 1.524m. 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 100m 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.
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 from Unarmored Movement (the amount of which we leave variable in order to determine the least amount need to break the world record). 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 use your free object interaction in order to poke yourself with a needle. :-)
The Monk's Unarmored Movement seems to be needed, though.
If we do not want to use any background magic at all, we could do a similar build as the preceding one where instead of Monk we take two Rogue Levels for Cunning Action, which has the same effect as Step of the Wind. However, we do not gain Unarmored Movement. In this case, the same arguments as above imply that we can run 100m in
$$6+\frac{100-44\cdot1.524}{\frac{33\cdot1.524}{6}}\approx9.93$$
seconds, which is not quite fast enough.