We consider the evolution in full general relativity of a family of linearly unstable isolated spherical neutron stars under the effects of very small perturbations as induced by the truncation error. Using a simple ideal-fluid equation of state, we find that this system exhibits a type I critical behaviour, thus confirming the conclusions reached by Liebling et al (2010 arXiv:1001.0575v1) for rotating magnetized stars. Exploiting the relative simplicity of our system, we are able to carry out a more in-depth study providing solid evidence of the criticality of this phenomenon and also to give a simple interpretation of the putative critical solution as a spherical solution with the unstable mode being the fundamental F-mode. Hence for any choice of the polytropic constant, the critical solution will distinguish the set of subcritical models migrating to the stable branch of the models of equilibrium from the set of subcritical models collapsing to a black hole. Finally, we study how the dynamics changes when the numerical perturbation is replaced by a finite-size, resolution-independent velocity perturbation and show that in such cases a nearly critical solution can be changed into either a sub- or supercritical one. The work reported here also lays the basis for the analysis carried in a companion paper, where the critical behaviour in the head-on collision of two neutron stars is instead considered (Kellerman et al 2010 Class. Quantum Grav. 27 235016).