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 -10,270 Å at a spatial resolution of 0 ′′ .05 -0 ′′ .1 and a spectral resolving power of λ/∆λ ≈ 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 o .In addition to the broad emission lines detected by spectropolarimetry, the hot spot shows the "little blue bump" in the 2000 -4000 Å 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 ∼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 towards 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, dominated by an old (≥ 2 x 10 9 years) stellar population, and includes a nuclear stellar cluster which is ∼200 pc in extent.