Bear with me as I'm a philosophy (not math) student. First some philosophical background, and then the math question.
One philosophical view is that physical space is composed of infinitely many points of zero volume. But the question arises, how can points of zero volume add up to a non-zero finite volume? Even with an infinite number of points this seems impossible--from nothing you get nothing, no matter how many times you add nothing.
The response to this question is typically an appeal to measure theory in mathematics. While the Lebesgue measure of a set containing a single point, e.g. $[1,1]$ is $0$, the Lebesgue measure of the set of points $[0,1]$ is non-zero. Hence we have what is in some sense an addition of $0$'s equaling something nonzero. Thus if it is the case that measure theory correctly describes physical space, we have an answer to our original question.
Now here is my question to you guys. Are there any alternatives to measure theory which yield different results? That is, are there any instances of mathematics where the sum of an actually infinite amount of zeros (this rules out limits and sums defined in terms of limits) is $0$? One candidate to me seemed to be the hyperreal numbers from non-standard analysis. While I do not know how to formalize an "infinite sum" in non-standard analysis (e.g. is there a sort of measure theory equivalent in non-standard analysis?) but I can do something similar--in non-standard analysis, $0$ times infinity is always zero (right?). Whether this calculation is a better analog to physical space than measure theory is something I am skeptical of, but nonetheless it would be interesting to hear from you mathematicians about sums of zeros in non-standard analysis or other alternative mathematical tools/theories.
Hyperreal infinitesimals do give you an alternative to Lebesgue measure when you want to determine the length of something (let's stick to length instead of volume to fix ideas, though there is no essential difference).
A short summary is that your idea of infinite sums can be realized in the following way. The interval $[0,1]$ is not viewed as a union of infinitely many points but rather is partitioned into an infinite number (more precisely, hyperfinite number) infinitesimal subintervals. Thus if you take a infinite hyperinteger $N$, the division points $\frac{i}{N}$ as $i$ "runs" from $0$ to $N$ give you a bunch of subintervals of infinitesimal length which sum up to the length of the interval $[0,1]$. The subinterval $[0, \frac{1}{2}]$ will only have half the partition intervals, and counting those you will only get half the length, as expected.
This does not provide all the details of the construction, but roughly speaking you can successfully implement the scheme that wants the "size" or "length" of something to be gotten essentially by counting.
In response to DRF's query, the point about hyperfinite partitions as that they contain enough information to be able to get the right size/length of sets, whereas for finite partitions there will always be a small error. This is similar to defining the derivative via the shadow/standard part function: one can actually get the value of the derivative out of the ratio of infinitesimals, rather than merely an approximation.
As for the mismatch (pointed out by DRF) between the quantum structure of physical space, on the one hand, and mathematical modeling thereof, on the other, this is correct. Furthermore, this is just as true with modeling using the real numbers as modeling using the hyperreal numbers.