If you allow the functions to be constant on some intervals, then there are some easy examples, and Ricky has provided one.

But if you rule that out, then there can be no examples, even with countably many functions. To see this,
suppose that $f_n$ is a list of countably many continuous functions which are never constant on an interval. Enumerate the pairs $(r,n)$ of
rational numbers $r$ and natural numbers $n$ in a countable list
$\langle (r_0,n_0), (r_1,n_1),\ldots\rangle$. Let $C_0$ be any closed
interval. If the closed interval $C_i$ is defined, consider
the function $f_{n_i}$ and the rational value $r_i$. Since $f_{n_i}$ is not constant value $r_i$ on $C_i$, we may
shrink the interval to $C_{i+1}\subset C_i$ such that
$f_{n_i}$ on $C_{i+1}$ is bounded away from $r_i$. By
compactness, there is some $x\in C_i$ for all $i$. Thus,
$f_n(x)$ is not $r$ for any rational number $r$.

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