Ever since he was a baby, he hasn't been able to sleep anywhere other than his own bed.
自分のベッドで寝れないと、旅行や用事で出かけた時、お母さん大変ですね！
ー Ever since he was a baby, he hasn't been able to sleep anywhere other than his own bed.
「赤ちゃんの時から、自分のベッド以外で寝られないんです。」
ever since ... 「…以来」という言い方です。
anywhere other than ... 「…以外のどこででも」となり、もし「自分の家以外のどこででも」なら anywhere other than his own house のようになります。
ご質問の日本語に近くなるように訳しましたが、私なら簡単に次のように言います。
ー He has only slept in his bed since he was a baby.
「赤ちゃんの時から自分のベッドでしか寝たことがない。」
自分のベッドでしか寝たことがないなら、ほかの場所では寝られないことになりますから。
ご参考まで！
Ever since he was a baby, he has only been able to sleep at home.
Ever since he was a baby, he hasn't been able to sleep outside of our home.
「赤ちゃんの時から家以外の場所では寝られない」は英語でこの文になります：
１）Ever since he was a baby, he has only been able to sleep at home.
２）Ever since he was a baby, he hasn't been able to sleep outside of our home.
気を使うポイントは、子供は男の子なら、「he」を使って、子供は女の子なら、「she」を使わなければなりません。
単語：
赤ちゃんの時 --> (the time when) he/she was a baby
から --> from/since/ever since
家以外の場所 --> outside of the home/away from home
寝られない --> can't sleep/isn't able to sleep/hasn't been able to sleep
カナダ 
アメリカ合衆国 
