Let's give some formulas for Achilles and the Tortoise, to see how the times when Achilles reaches a point where the Tortoise was previously at keeps shrinking and stays below a definite time (when their positions are tied) rather than getting big.

Say Achilles has position A(t) = t at time t and the Tortoise has position T(t) = 4 + t/100 at time t, so A(0) = 0 and T(0) = 4: at time 0, the tortoise is 4 units ahead (meters, feet, whatever).

Achilles reaches position 4 at time 4: A(4) = 4. And at that time T(4) = 4.04 > 4, so the tortoise is still ahead at time 4.

Achilles reaches position 4.04 at time 4.04: A(4.04) = 4.04. And at that time T(4.04) = 4.0404 > 4.04, so the tortoise is still ahead at time 4.04.

Achilles reaches position 4.0404 at time 4.0404: A(4.0404) = 4.0404. And at that time T(4.0404) = 4.040404 > 4.0404, so the tortoise is still ahead.

Achilles reaches position 4.040404 at time 4.040404: A(4.040404) = 4.040404. And at that time T(4.040404) = 4.04040404 > 4.040404, so the tortoise is still ahead.

Despite the tortoise always being ahead of Achilles at these times, notice the times we are working with are always remaining below 4.040404040404... < 4.05. So it's simply false that Achilles "never" catches the tortoise because such reasoning is just ignoring the actual passage of time by focusing on ever vanishingly small units of time. Once the time reaches 400/99 = 4.040404040404..., Achilles and the tortoise are at the same position: their positions are tied. And after this time Achilles gets ahead of the tortoise and remains ahead. Notice the time when Achilles and the tortoise are tied is the value of an infinite decimal 4.0404040404..., which can be thought of as a convergent infinite series 4 + .04 + .0004 + .000004 + ..., which is related to the answer by u/EquinoctialPie saying that the resolution of the paradox is the fact that infinite series can have finite values. One does not need other tools from calculus (like integrals), but perhaps the series can be viewed in different ways related to those other tools.

You can graph this: plot y = t and y = 4 + t/100. For very small positive t, we have t < 4 + t/100 (Achilles is behind the tortoise), but when t = 400/99 the two lines cross, and when t > 400/99 the first line is higher than the second line (Achilles is ahead of the tortoise).

In summary, there is no actual paradox if you pay attention to all times instead of getting fixated only on quite small times. The error here is analogous to people who mistakenly think an unending decimal like pi = 3.141592653589... is an "infinite number" because it has infinitely many digits: this confuses the number of digits with the numerical value of the decimal.

If your question is about how anything can move at all "by Newton's third law", please see https://physics.stackexchange.com/questions/45653/given-newtons-third-law-why-are-things-capable-of-moving