A de Bruijn sequence is a (typically) binary string of length $2^k$ which contains every binary string of length $k$ as a substring exactly once, if you allow it to wrap around. For example, 00010111 works for $k=3$.
I was wondering whether you could have one of infinite length, and brief consideration reveals that no, you can't; among other things, it would have to contain all possible binary sequences—themselves of infinite length—which would push it up to the cardinality of the continuum.
However, is it possible to make an optimal attempt at it? If we define optimal as the infinitely-long binary string for which the average distance in you need to look to first encounter any given substring is minimized, where 'average' is taken over all possible strings of some fixed length. It seems like you would also have to weight it so that shorter substrings are favored before longer ones, and repetitions of substrings of a given length avoided as much as possible.
If nothing else, I can reasonably say that it wouldn't start with 00000, but rather with something like 0110001... perhaps gluing de Bruijn sequences together for increasing $k$ is the best we can do?