In the numeration system with base $5$, counting is as follows: $1, 2, 3, 4, 10, 11, 12, 13, 14, 20,\ldots$.
The number whose description in the decimal system is $69$, when described in the base $5$ system, is a number with:
$\textbf{(A)}\ \text{two consecutive digits} \qquad\textbf{(B)}\ \text{two non-consecutive digits} \qquad \\ \textbf{(C)}\ \text{three consecutive digits} \qquad\textbf{(D)}\ \text{three non-consecutive digits} \qquad \\ \textbf{(E)}\ \text{four digits}$
Since $25<69<125$, divide $69$ by $25$.  The quotient is $2$ and the remainder is $19$, so rewrite the number as
\[69 = 2 \cdot 25 + 19\]
Similarly, dividing $19$ by $5$ results in quotient of $3$ and remainder of $4$, so rewrite the number as
\[69 = 2 \cdot 25 + 3 \cdot 5 + 4 \cdot 1\]
Thus, the number in base $5$ can be written as $234_5$, so the answer is $\boxed{\textbf{(C)}}$