Is a lack of any barriers to entry a necessary condition for ensuring pareto optimality is reached? I understand that a competitive market is necessary for a market to equilibriate to a pareto optimal state, and that things like economies of scale, network effects, government restrictions, and government imposed costs can prevent reaching pareto optimality. But do other barriers like startup capital also prevent reaching pareto optimality? I would think the answer is no, since something like sunk costs can be amortized over all products sold by a market actor. Is this correct, or am I missing something?
Let us quickly define a competitive market. A competitive market satisfies the following:

1) All firms sell an identical product

2) All firms and buyers are price takers (have no control over price)

3) Buyers and firms have complete information about the market

4) The market is characterized by free entry (no barriers to entry)

If this is the market structure, we can have the existence of pareto optimality, so I am assuming this is the competitive market to which you are referring in your question. What happens when we try to relax assumption 4?

We can show this using your example of startup capital: 

Suppose we have a competitive market and the amount of startup capital needed to enter this market is $A$. When deciding to enter this market the firm would subtract $A$ from the long term profit possibility $\Pi$ (because it would be another cost). By definition $A\geq 0$. Because the situation where $A=0$ is trivial (because that means there is no startup capital needed), we know that $A&gt;0$. Suppose at a point in time there is opportunity for profit in this market. As long as $\Pi\geq 0$ firms will decide to enter the market, but if $\Pi &lt;0$, a firm would decide not to enter the market. However, note that firms who are already in the market are earning positive profit. This is a contradiction because if positive profits persist, we do not have a competitive market. 

As you noted in your question, a competitive market is a necessary condition for the existence of pareto optimality. If the nonexistence of barriers to entry is a necessary condition for a competitive market, nonexistence of barriers to entry is also a necessary condition for pareto optimality.