  Strongly bounded groups are those groups for which every action by isometries
on a metric space has orbits of finite diameter. Many groups have been shown to
have this property, and all the known infinite examples so far have cardinality
at least $2^{\aleph_0}$. We produce examples of strongly bounded groups of many
cardinalities, including $\aleph_1$, answering a question of Yves de Cornulier
[4]. In fact, any infinite group embeds as a subgroup of a strongly bounded
group which is, at most, two cardinalities larger.
