Do neutrons in an atomic nucleus exert strong force on each other? If they do, then why do we never see clumps of neutrons assembled into  "atoms" without protons? Or are neurtons mutually repelled by the strong force? 
 A: Yes, neutrons are influenced by the strong nuclear force inside nuclei. If they did not then how could they be bound in nuclei at all?
Nuclei cannot be built solely of neutrons because they would decay via weak interactions into protons. It is only when the weak decay is inhibited by the presence of protons already occupying the low-lying energy states that neutrons could potentially decay into (protons are fermions and no two identical fermions can occupy the same quantum state), that stability can be reached.
A: The description of the neutron-neutron interaction at low energy can be summarized using a scattering length.  The relationship between the scattering length, the quantum-mechanical phase shift between scatterers, and the "apparent size" of an object requires some careful thought about quantum mechanics.  However for our purposes here we can say that a large scattering length corresponds to a hefty interaction.
According to some plausible-looking presentation, some recent values for the nucleon-nucleon scattering lengths $a$ and effective interaction ranges $r$ are
$$
\begin{array}{ccc}
NN & a\,\mathrm{ (fm)} & r \,\mathrm{ (fm)} \\
\hline
nn & -18.9 \pm 0.4 & 2.75 \pm 0.11 \\
np & -23.740 \pm 0.020 & 2.77 \pm 0.05 \\
pp & -17.3 \pm 0.4 & 2.85 \pm 0.04
\end{array}
$$
From this you can see that the neutron-neutron interaction is slightly stronger than the proton-proton interaction.  From the uncertainties you can also see that neutrons are slipperier to work with than protons.  (The $np$ data are mostly from deuterium.)
From the perspective of the strong force the proton and neutron are the same particle: the nucleon.  If the di-neutron were stable, we would expect also to also to find stable di-protons and a bound spin-zero excitation in deuterium; none of those objects exist in nature, but that is not evidence against the proton-proton strong interaction.
In heavier nuclei, a nice handwaving model is that the nucleons are all paired up, in the same way that electrons pair up in atomic orbitals.  In "odd-odd" nuclei like postassium-40 (19 protons, 21 neutrons), the last pair is mixed, $np$.  The nucleus doesn't like this, and the mixed pair beta-decays to $pp$ (calcium-40, 90% of decays) or $nn$ (argon-40 10% of decays) with comparable probability.  That suggests that the neutron-neutron interaction, like all the nucleon-nucleon interactions, can be either attractive or repulsive depending on circumstances; in the mass-40 system apparently $np$ is less attractive that $nn$.
A: 
Do neutrons in an atomic nucleus exert strong force on each other?

No. There are no nuclei that consist of neutrons alone. You need protons to "stick" the neutrons together, and you need neutrons to "stick" the protons together. Have a look at the neutron/proton ratio. NB: there's some ambiguity about the strong force, see the Wikipedia strong interaction article. I tend to think of the strong force as the force between the quarks of a neutron or a proton, whilst the residual strong force is the force between protons and neutrons. This is also called the nuclear force. 

If they do,then why do we never see clumps of neutrons assembled into "atoms" without protons? Or are neutrons mutually repelled by the strong force?

I don't think they're repelled so much as they aren't attracted. Check out things like the dineutron and the trineutron on Wikipedia. 
"The dineutron, containing two neutrons was unambiguously observed in the decay of beryllium-16, in 2012 by researchers at Michigan State University.[6][7] It is not a bound particle, but had been proposed as an extremely short-lived state produced by nuclear reactions involving tritium..." 
"A trineutron state consisting of three bound neutrons has not been detected, and is not expected to exist even for a short time."
Think of a nucleus as something akin to magnetic stix and balls. The balls aren't magnetic. 

A: There is such a theoretical element with atomic number zero.
http://periodictableofelements.wikia.com/wiki/Neutronium
