Yes. Middle C is C4 in scientific pitch notation, and the baritone is tuned D3G3B3E4. So middle C can be found on the first fret of the B string/third string from the top. It can also be found on the fifth fret of the G string/second string from the top, and the tenth fret of the first string.
However, I realize you might be asking if there is a "middle pitch" of the range of the instrument. Stringed instruments are not truly chromatic from one string to another as a piano is (i.e there are multiple of the same notes), but a normal baritone range with 15 frets is from D3 to G5, which puts this range at 30 half steps. The middle of this range is 15 half steps above D3, which is about E4. So, E4 is the middle of the range, and this note can be found on the last string plucked open or the last string, first fret. It can also be found on the B string, fifth fret, and the G string, ninth fret, and the D string, 13th fret.