Fucoidan, a sulphated polysaccharide from brown seaweed composed of several monosaccharides, has been stated to have several bioactive properties such as antioxidant, antiviral, anticancer, antithrombic, anti-inflammatory, and immunomodulatory effects. This paper provides research findings on green extraction methods, structural analysis of fucoidan, and its associated bioactivities. Fucoidans from brown seaweeds, Fucus vesiculosus and Ascophyllum nodosum, were extracted using green solvents such as citric acid (CA) followed by MWCO (molecular weight cut-off) filtration to obtain high-purity polysaccharides. The presence of functional groups typical to fucoidans, namely, fucose, sulfate, and glycosidic bonds, in the extracts were confirmed through the data obtained from FTIR (Fourier-transform infrared spectroscopy), TGA (thermogravimetric analysis), DSC (differential scanning calorimetry), and solid-state CP–MAS (cross-polarization magic angle spinning) analysis. The MWCO analysis identified that the >300 kDa fraction can have better content of fucoidan (FV-CA 79.16%, FV-HCl 63.59%, AN-CA 79.21%, AN-HCl 80.70%) than the conventional extraction process. Furthermore, the >300 kDa fraction showed significantly higher antioxidant activities compared to crude fucoidan extracts. Crude fucoidan extracts showed significant inhibition of cell viability in human lung (A459 lung carcinoma cells) and colorectal adenocarcinoma (Caco-2) cells at higher concentrations. The fucoidan extracted with green solvents and avoiding alcohol-based precipitation has substantial antioxidant/antitumor action, so, due to this activity, it can be employed as functional foods in food applications.