We present the first long-slit spectra of the Seyfert 2 galaxy NGC 1068 obtained by the Space Telescope Imaging Spectrograph (STIS); the spectra cover the wavelength range 1150-10270 Angstrom at a spatial resolution of 0 ".05-0 ".1 and a spectral resolving power of lambda/Delta lambda approximate to 1000. In this first paper, we concentrate on the far-UV to near-IR continuum emission from the continuum "hot spot" and surrounding regions extending out to +/-6 ".0 (+/-432 pc) at a position angle of 202 degrees. In addition to the broad emission lines detected by spectropolarimetry, the hot spot shows the "little blue bump" in the 2000-4000 Angstrom range, which is due to Fe II and Balmer continuum emission. The continuum shape of the hot spot is indistinguishable from that of NGC 4151 and other Seyfert 1 galaxies. Thus, the hot spot is reflected emission from the hidden nucleus, due to electron scattering (as opposed to wavelength-dependent dust scattering). The hot spot is similar to 0 ".3 in extent and accounts for 20% of the scattered light in the inner 500 pc. We are able to deconvolve the extended continuum emission in this region into two components: electron-scattered light from the hidden nucleus (which dominates in the UV) and stellar light (which dominates in the optical and near-IR). The scattered light is heavily concentrated toward the hot spot, is stronger in the northeast, and is enhanced in regions of strong narrow-line emission. The stellar component is more extended, concentrated southwest of the hot spot, is dominated by an old (greater than or equal to 2 x 10(9) yr) stellar population, and includes a nuclear stellar cluster that is similar to 200 pc in extent.