[verb て-form] + ばかりはいられない [verb て-form] + ばかりもいられない
てばかりはいられない means "I can't just keep doing this": the situation won't let you go on doing only one thing. It is てばかりいる, "do nothing but", with いる in the negative potential いられない, and it follows the て-form of a verb.
The verb is usually a passive or unproductive state – crying, worrying, feeling down, resting, relying on others – and the next sentence often says what has to happen instead. It is said to yourself as resolve, or to someone else as a push.
These go beyond the basic form.
いられない becomes いられません.