Yes, neutron repels each other.
Neutrons are electrically neutral particles and do not have any net charge on the neutron.
Despite, the net charge is zero, but still, neutron repels another neutron.
Quantum physics can explain the cause of this repulsion.
In classical physics, we have told that neutron is neutral and does not repel each other, and classical physics does not discuss anything about the distance between neutrons.
Let consider, some example before diving deeper to answer the question.
I hope you will be familiar with the beta emission from the nucleus of a radioactive element. Beta particle, which is a negatively charged particles having a magnitude equal to the electron. How the beta emission occurs from the nucleus? It is possible when a neutron transformer into a proton, then beta emission occurs to obey the law of conservation of charges.
Now let’s, consider one more example; have you ever think about the molecule of water, which is neutral electrically but still water has positive and negative charges at the two ends, and that’s why water is the example of a dipole molecule. I hope you will know about the capacitor which we use in almost every electronic circuit. If we take the capacitor as a closed system, then we can say that there is no net charge on the capacitor. But if you take the two plates of the capacitor individually, then you will see a positive charge on one plate and a negative one on another. And as a result, the capacitor behaves like a dipole.
Before going into detail, I will discuss here the structure of the neutron. The neutron is made of two down quarks and one up quark.
Up quark has an electric charge of +2/3e, while down quark has -1/3e. When these three quarks combine and make a neutron, that neutron gets a net charge equal to zero. And due to this reason, classical physics claims that neutron does not repel another neutron. But this is not true according to quantum physics.
When two neutrons are brought close to each other, then the quarks rearrange themselves, and the opposite charges appear near to each other, and these quarks will attract each other.
In the subject of quantum physics, neutrons are called fermions, and neutrons have a spin of ½ and obey Pauli exclusion principle. According to the Pauli Exclusion principle, two neutrons can not fill the same space at the same time, and their wave functions are antisymmetric. The wave function produces: (1) a spatial part, (2) a spin part, and (3) an isospin part. When the wave function of two neutron’s overlap, they feel a strong repulsive force.