Electrostatic interaction between protein and polysaccharides could influence structured liquid oil stability when emulsification is used for this purpose. The objective of this work was to structure sunflower oil forming emulsions and High Internal Phase Emulsions (HIPEs) using pea protein (PP) and xanthan gum (XG) as a stabilizer, promoting or not their electrostatic attraction. The 60/40 oil-in-water emulsions were made varying the pH (3, 5, and 7) and PP:XG ratio (4:1, 8:1, and 12:1). To form HIPEs, samples were oven-dried and homogenized. The higher the pH, the smaller the droplet size (Emulsions: 15.60–43.96 µm and HIPEs: 8.74–20.38 µm) and the oil release after 9 weeks of storage at 5 °C and 25 °C (oil loss < 8%). All systems had weak gel-like behavior, however, the values of viscoelastic properties (G′ and G″) increased with the increment of PP:XG ratio. Stable emulsions were obtained at pHs 5 and 7 in all PP:XG ratios, and at pH 3 in the ratio 4:1. Stable HIPEs were obtained at pH 7 in the ratios PP:XG 4:1, 8:1, and 12:1, and at pH 5 at PP:XG ratio 4:1. All these systems presented different characteristics that could be exploited for their application as fat substitutes.