DFA for w|w has a length multiple of 2 or 3
Creating a state machine for each and combining the two is not a bad idea. To create a state machine accepting words having length N you can have each state represent the current length mod N.

After creating the two machines you can add a state having two epsilon transitions to the starting states of both machines. This gives a NFA accepting your language.

In a final stage you use the subset construction algorithm to convert the NFA you got into a DFA.