  In 1922, Mordell conjectured that the set of rational points on a smooth
curve $C$ over $\mathbb{Q}$ with genus $g \ge 2$ is finite. This has been
proved by Faltings in 1983. However, Coleman determined in 1985 an upper bound
of $#C(\mathbb{Q})$ by following Chabauty's approach which considers the
special case when the Jacobian variety of $C$ has Mordell-Weil rank $< g$. In
2006, Stoll improved the Coleman's bound. Balakrishnan with her co-authors in
[1] implemented the Chabauty-Coleman method to compute the rational points of
genus $3$ hyperelliptic curves. Then, Hashimoto and Morrison [8] did the same
work for Picard curves. But it happens that this work has not yet been done for
all genus 3 curves. In this paper, we describe an algorithm to compute the
complete set of rational points $C(\mathbb{Q})$ for any genus $3$ curve
$C/\mathbb{Q}$ that is a degree-$2$ cover of a genus $1$ curve whose Jacobian
has rank $0$. We implemented this algorithm in Magma, and we ran it on
approximately $40, 000$ curves selected from databases of plane quartics and
genus $3$ hyperellitic curves. We discuss some interesting examples, and we
exhibit curves for which the number of rational points meets the Stoll's bound
