Can we add an uncountable number of positive elements, and can this sum be finite?

Suppose $\{s_i : i\in\mathcal I\}$ is a family of positive numbers.$^\dagger$ We can define $$ \sum_{i\in\mathcal I} s_i = \sup\left\{ \sum_{i\in\mathcal I_0} s_i : \mathcal I_0 \subseteq \mathcal I\ \&\ \mathcal I_0 \text{ is finite.} \right\} $$ (If both positive and negative numbers are involved, then we have to talk about a limit rather than about a supremum, and then the definition is more complicated and we have questions of conditional convergence and rearrangements.)

Now consider \begin{align} & \{i\in\mathcal I : s_i \ge 1\} \\[4pt] & \{i\in\mathcal I : 1/2 \le s_i < 1 \} \\[4pt] & \{i\in\mathcal I : 1/3 \le s_i < 1/2 \} \\[4pt] & \{i\in\mathcal I : 1/4 \le s_i < 1/3 \} \\[4pt] & \quad \quad \quad \vdots \end{align} If one of these sets is infinite, then $\sum_{i\in\mathcal I} s_i=\infty$. But if all are finite, then $\mathcal I$ is at most countably infinite.

Thus the sum of uncountably many positive numbers is infinite.

I don't know whether by some arguments about rearrangements one could somehow have some sensible definition of a sum of numbers not all having the same sign that could give us a somehow well defined sum of uncountably many numbers and get a finite number.

$^\dagger$ In the initial edition of this answer, I said "Let $S$ be a set of positive numbers and then went on to say $$ \sum S = \left\{ \sum S_0 : S_0\subseteq S\ \&\ S_0\text{ is finite.} \right\} $$ However, Dustan Levenstein pointed out in comments that "this definition fails to allow for the same number to occur twice in a sum". Rather than "twice", I'd say "more than once", since a number might even occur an uncountably infinite number of times.

We have the next proposition

Proposition 1. Let $X$ be an at most countable set, and let $f\colon X\to\mathbf R$ be a function. Then the series $\sum_{x\in X} f(x)$ is absolutely convergent if and only if $$\sup\left\{\sum_{x\in A}|f(x)|:A\subseteq X, A\text{ finite}\right\}<\infty.$$

Inspired by this proposition, we may now define the concept of an absolutely convergent series even when the set $X$ could be uncountable.

Definition 2. Let $X$ be a set (which could be uncountable), and let $f\colon X\to\mathbf R$ be a function. We say that the series $\sum_{x\in X} f(x)$ is absolutely convergent if and only if $$\sup\left\{\sum_{x\in A}|f(x)|:A\subseteq X, A\text{ finite}\right\}<\infty.$$

Note that we have not yet said what the series $\sum_{x\in X} f(x)$ is equal to. This shall be accomplished by the following proposition.

Proposition 3. Let $X$ be a set (which could be uncountable), and let $f\colon X\to\mathbf R$ be a function such that the series $\sum_{x\in X} f(x)$ is absolutely convergent. Then the set $\{x\in X:f(x)\ne0\}$ is at most countable.

Because of this, we can define the value of $\sum_{x\in X} f(x)$ for any absolutely convergent series on an uncountable set $X$ by the formula $$\sum_{x\in X} f(x):=\sum_{x\in X:f(x)\ne0} f(x),$$ since we have replaced a sum on an uncountable set $X$ by a sum on the countable set $\{x\in X:f(x)\ne0\}$. (Note that if the former sum is absolutely convergent, then the latter one is also.) Note also that this definition is consistent with the definitions for series on countable sets.

Remark. The definition of series on countable sets that are use is

Definition 4. Let $X$ be a countable set, and let $f\colon X\to\mathbf R$ be a function. We say that the series $\sum_{x\in X}f(x)$ is absolutely convergent iff for some bijection $g\colon\mathbf N\to X$, the sum $\sum_{n=0}^\infty f(g(n))$ is absolutely convergent. We then define the sum of $\sum_{x\in X}f(x)$ by the formula $$\sum_{x\in X}f(x)=\sum_{n=0}^\infty f(g(n)).$$

Let $H$ be a positive unlimited integer of nonstandard analysis. Then, for example, the sum

$$ \sum_{n=1}^H n = \frac{H(H+1)}{2}$$

is a sum of uncountably many positive numbers... but it's a hyperfinite nonstandard sum, so it exists by the usual methods of nonstandard analysis. The sum is unlimited, though. Other sums can be finite: e.g.

$$ \sum_{n=1}^H \frac{1}{n!}$$

is a finite nonstandard real number that is infinitesimally close to $e$.

That said, IMO, thinking of hyperfinite sums from nonstandard analysis as being sums of uncountably many elements isn't a particularly fruitful line of thought. (also, the sum only works for internal sequences of elements anyways; you can't take an arbitrary uncountable collection)

I bring this up mainly to show that uncountable sums can make sense in some contexts, even if you can't really do much in a standard setting. Each summation operator one might define can have its own sorts of pecularities.