5-state DFA that recognizes only the string 00001111
[If I understand the question correctly, this only relates to strings of length 8, i.e. for an automata $A$ we only look at $L(A) \cap \Sigma^8$]

There is a slight variance in interpretation of what a DFA is that plays a role here.

You can construct a DFA with 5 states that accepts $00001111$, but it is not complete, meaning not every state has an exiting transition for every symbol of the alphabet. This quesiton addresses this point in more details.

What this difference in notation allows you to do is skip the sink-node:


If the input string is of length 8, this automaton accepts only $00001111$.