We study two models of scalar dark matter from "large" electroweak multiplets with isospin 5/2 (n=6 members) and 7/2 (n=8), whose scalar potentials preserve a Z2 symmetry. Because of large annihilation cross sections due to electroweak interactions, these scalars can constitute all the dark matter only for masses in the multi-TeV range. For such high masses, Sommerfeld enhancement and coannihilations play important roles in the dark matter relic abundance calculation, reducing the upper bound on the large multiplet's mass by almost a factor of 2. We determine the allowed parameter ranges including both of these effects and show that these models are as yet unconstrained by dark matter direct detection experiments, but will be probed by currently running and proposed future experiments. We also show that a Landau pole appears in these models at energy scales below 109 GeV, indicating the presence of additional new physics below that scale.