There are some things that should be addressed first.
This is a condition at which the forward reaction rate is equal to the backward reaction rate for a reversible reaction.
In this case, $k_f.A=k_r.B$, at the equilibrium, if the reactions are first-order. If $k_f>k_r$, then the concentration of B would be higher than the concentration of A, at equilibrium point.
It is a state of the system at which the overall concentration of all the components (reactants and products) remain constant with time, or oscillate at a constant amplitude and frequency. Mathematically, the differential equations that denote the system (as mentioned by Dexter) would equate to zero.
If this condition applies only for one or more but not all the components of the system, then it is said to be in a Quasi-Steady-State.
If there are only two species which inter-convert but do not get formed or degraded, then steady state would be same as the chemical equilibrium. However, it is to be noted that chemical-equilibrium is a property of a reversible reaction whereas steady state is the property of the system.
Coming to your main question, I do not know what your teacher really had in mind, but this is my explanation.
According to the above definitions, you cannot say that the cell is in equilibrium because the cell is a system and not a reaction. The usage of the term "equilibrium" varies between different fields of study and physicists do use steady-state and equilibrium interchangeably. However, this is not the practice among chemists and biochemists.
Having said that, I would say that a cell does attain a steady state. I think what your teacher wanted to imply was that a cell constantly responds to the external stimuli (which is randomly changing) and therefore never remains at any given steady state for a long time.
Can the cells not exist at static equilibrium?
No, they don't. In fact the term "static equilibrium" in never used in chemistry because it never really happens. At a static equilibrium all the rates have to be zero, whereas in a dynamic equilibrium the rates have to balance each other out. Static equilibrium usually applies only for macroscopic objects (which do not include molecules).
A note on stochasticity: Chemical reactions follow probabilistic kinetics i.e. the reactions occur with some probabilities, instead of fixed rates (these probabilities can be unequal for different reactions). Chemical systems can be approximated by deterministic models at the thermodynamic-limit which reflects the "average" behaviour of the system.
when something is at dynamic equilibrium, why does this mean that the
concentrations of reactants and products are not necessarily equal?
See the first section. Similar logic applies for steady-state. The steady state concentrations depend on the kinetic parameters.