Strings of infinite length?
Here is a simpler example. Consider the set $\mathbb{N}$ of natural numbers:
$$
\mathbb{N} = \{0, 1, 2, \ldots\}.
$$
The set $\mathbb{N}$ is infinite, but each number is finite.

In fact, your situation is very similar, since
$$
\Sigma^* = \bigcup_{n \in \mathbb{N}} \Sigma^n.
$$
Each of the sets $\Sigma^n$ is finite, yet the set $\Sigma^*$ is infinite.

Each word in each $\Sigma^n$ is finite, so each word in $\Sigma^*$ is finite.



Here is a related point.

Words in $\Sigma^n$ have bounded length, but the length of words in $\Sigma^*$ is unbounded. Yet the length of each word itself is finite.

Similarly, each number in $\mathbb{N}$ is finite, yet $\sup \mathbb{N} = \infty$.



Finally, let me say that words of infinite length are quite useful. The set $\Sigma^\omega$, known as the set of $\omega$-words, consists of all "one-way" infinite words. A word in $\Sigma^\omega$ is a mapping $w\colon \omega \to \Sigma$ ($\omega$ is another symbol for $\mathbb{N}$). We think of it as given by symbols $w_n \in \Sigma$ for all $n \in \mathbb{N}$.

Two-way infinite words form $\Sigma^\zeta$, where $\zeta$ is the same as the set $\mathbb{Z}$ of all integers.

More generally, you can consider $\Sigma^\alpha$ for every linear order $\alpha$.