Parliament Street, in Exeter, narrows to 25" and dates from the 1300s. Ebenezer Place, in Scotland, runs along the short side of the hotel shown, for a total length of 2 meters.
Credit here, where there are also pix of streets that are the longest, crookedest, oldest, widest, steepest...
Update: see the note by Megan in the Comments section re a possibly narrower street in Mexico, tho it may raise the question of the distinction (if there is one?) between an "alley" and a "street."