How to compute transfer function $G(s) = \exp \left( - \sqrt{s} \right)$ in Matlab / Simulink?
The inverse Laplace transform of $G(s)=e^{-\sqrt{s}}$ can be computed analytically:
$$g(t)=\mathcal{L}^{-1}\left\{e^{-\sqrt{s}}\right\}=\frac{1}{2t\sqrt{\pi t}}e^{-\frac{1}{4t}},\qquad t&gt;0\tag{1}$$
[I've cross-checked this result by looking it up in one of my old math books.]
So now you have an analytical expression for the impulse response $g(t)$.
The step response can also be computed analytically. It is given by
$$s(t)=u(t)\int_0^t g(\tau)d\tau=\textrm{erfc}\left(\frac{1}{2\sqrt{t}}\right)u(t)\tag{2}$$
where $u(t)$ is the unit step function, and $\textrm{erfc}(\cdot)$ is the complementary error function.
The result $(2)$ was obtained with a little help from WolframAlpha.
And that's how the impulse response and step response look like: