A solid torus should work. Choose cylindrical coordinates $(r,\theta, \lambda)$, $0\leq r \leq r_0 < \pi/2, 0\leq \theta \leq 2\pi, 0\leq \lambda \leq l$, where we equate $(r,\theta,0)\sim (r,\theta, l)$ and $(0,\theta, \lambda)\sim (0,0,\lambda)$. Put a Riemannian metric on this solid torus of the form $dr^2+ f(r)^2 d\theta^2 + g(r)^2 d\lambda^2$, and $f(r)=\sin(r), g(r)=\cosh(\epsilon r)$, where $0 < \epsilon$ is small.
The sectional curvatures of such a metric are computed in Lemma 2.3 of this paper as
$$K_{\theta\lambda}=-\frac{f'g'}{fg},\ K_{r\theta}=-\frac{f''}{f},\
K_{r\lambda}=-\frac{g''}{g}$$
and the mean curvature of the level $r$ torus is
$$\frac12(\frac{f'}{f}+\frac{g'}{g}).$$
From the sectional curvatures, we get the scalar
curvature as
$$R= -2 (\epsilon \cos(r) \sinh(r) -\sin(r) \cosh(\epsilon r) +\epsilon^2 \sin(r) \cosh(\epsilon r))/fg,$$
and mean curvature of the level surface at height $r$ as
$$ \frac12(\cos(r)\cosh(\epsilon r) +\epsilon\sin(r) \sinh(\epsilon r))/fg.$$
We see that the level surfaces $r=c$ are mean convex tori, and the scalar curvature is positive for $\epsilon$ and $r_0$ small. Hence this metric contains no closed minimal surface: the maximal $r$ value for such a surface would be tangent to a level surface which is mean convex, contradicting the maximum principle.
Here's how I found this metric: given your criteria, the double of the manifold admits a metric with positive scalar curvature (this is a trick of Hubert Bray; cf. also Pengzi Miao). Such manifolds are connect sums of space forms and $S^2\times S^1$. The reflection symmetry quotient then gives a handlebody (a space form with non-trivial fundamental group cannot admit a reflection symmetry with fixed set a surface). Hence the simplest non-trivial case is a solid torus. The above metrics are invariant under the $S^1\times S^1$ action on a solid torus (a ``double-warped product"). The $(r,\theta)$ slice is a spherical cap, and the $(r,\lambda)$ slices are scaled hyperbolic metrics on a cyinder. Then we adjust $\epsilon$ and $r_0$ to make the scalar curvature positive, the positive sectional curvature of the spherical cap dominating the negative curvature of the hyperbolic annulus. I suspect one can use the techniques in Codá-Marques' paper to realize any handlebody with such properties.
