Hmmm, why do people always insist on a spherical mass and the $\frac{1}{r^2}$ law?
Gravity and electrostatics are very very similar. For electrostatics, Maxwell's first equation is
$$\frac{Q}{\varepsilon_0}=\int_A\vec{E}d\vec{A}$$
Which means: If you have a closed surface and sum up the product of surface times perpendicular E-field, you get the charge enclosed divided by $\varepsilon_0$. In case of a spherical charge distribution, you can put a sphere of radius r and surface $4\pi r^2$ around it. The field is constant and perpendicular at each point of the surface, so the equation becomes simple and gives a well known result:
$$\frac{Q}{\varepsilon_0}=4\pi r^2 E \quad \Rightarrow\quad E=\frac{Q}{4\pi\varepsilon_0 r^2}$$
Now, if there's an infinite plate with charge Q, you would place two planes on both sides of it as surface. This time, you get
$$\frac{Q}{\varepsilon_0}=2A E \quad \Rightarrow\quad E=\frac{Q}{2A \varepsilon_0}=\frac{\rho}{2 \varepsilon_0}$$
Here, $\rho$ ist the charge per surface.
The result is remarkable because first, the field is parallel everywhere and points to the plate and second, it does not depend on distance!
Back to gravity, we know a sphere gives
$$a=G\frac{M}{r^2}$$
Comparing this to the formula for a spherical charge gives
$$G\equiv\frac{1}{4\pi\varepsilon_0} \quad\Rightarrow\quad2\pi G\equiv\frac{1}{2\varepsilon_0}$$
which can be used for the formula of the plate. So, a plate with given weight per surface $\rho$ generates a constant (!) gravitational field of
$$a=2\pi G\rho$$
For a=9.81m/s², one gets $\rho=23.4\cdot10^9kg/m^2$.
Osmium has a density of $22.6\cdot10^3kg/m^3$, so the plate would have a thickness of 1000km.
Neutronium, having a density of $3\cdot10^{17}kg/m^3$ would only require a layer of 0.08µm.
BUT:
The command module of the apollo missions had a diameter of 3.9m, or a base area of 12m². This would require a "neutronium carpet" with a weight of 280,800,000t (For comparison: Entire Saturn V: 3,000t; cheops pyramid: 6,500,000t). So, no way to lift/move this, and even no way to support it while the rocket is on the ground.
Further more, the field of this plane can't be considered constant,since the dimension of the module is large compared to the diameter. A space ship shaped more like a flying saucer would be better.
And as said in the other answers: Neutronium only exists under high pressure. Under "normal" conditions, it would for sure expand (explode?)to a more common type of matter.