We describe here the fabrication, characterization, and properties of tough bioplastics made of a babassu oil-based acrylic polymer (PBBM), hemicellulose xylan grafted with PBBM chains, and carnauba wax (CW). The plastic was primarily designed to obtain bioderived materials that can replace low-density polyethylene (LDPE) in certain food packaging applications. To obtain plastic, the radical polymerization of an original babassu oil-based acrylic monomer (BBM) in the presence of xylan macromolecules modified with maleic anhydride (X-MA) was conducted. The polymerization resulted in a material (PBBM-X) mostly consisting of highly branched PBBM/X-MA macromolecules. PBBM-X has a glass transition of 42 °C, a storage modulus of 130 MPa (at 25 °C, RT), and a Young’s modulus of 30 MPa at RT. To increase the moduli, we blended PBBM-X with carnauba wax, a natural material with a high modulus and a melting temperature of ~80 °C. It was found that PBBM-X is compatible with the wax, as evidenced by the alternation of the material’s thermal transitions and the co-crystallization of BBM side alkyl fragments with CW. As a result, the PBBM-X/CW blend containing 40% of the wax had a storage modulus of 475 MPa (RT) and a Young’s modulus of 248 MPa (RT), which is close to that of LDPE. As polyethylene, the PBBM-X and PBBM-X/CW bioplastics have the typical stress-strain behavior demonstrated by ductile (tough) plastics. However, the bioplastic’s yield strength and elongation-at-yield are considerably lower than those of LDPE. We evaluated the moisture barrier properties of the PBBM-X/(40%)CW material and found that the bioplastic’s water vapor permeability (WVP) is quite close to that of LDPE. Our bioderived material demonstrates a WVP that is comparable to polyethylene terephthalate and lower than the WVP of nylon and polystyrene. Taking into account the obtained results, the fabricated materials can be considered as polyethylene alternatives to provide sustainability in plastics production in the packaging areas where LDPE currently dominates.