Is five an unlucky number?

Is it true that five is considered an unlucky number because the word for "five" sounds just like a word meaning "mistake"?

No.

I am a Japanese but never have been heard of that. Four is considered to be unlucky number, so many hotel tend to avoid it for the room numbers, but it is not considered seriously, in general. It might be noted for gifts for some ceremonies such as wedding, but other than that, many Japanese don't mind about it.

In Chinese history, No. 5 has a wide range of use in classifying things with similar characters, which makes things easy to be remembered. Here are some examples:
1. Five Flavors: sweet, sour, bitter, pungent, salty
2. Five Classics in Ancient China: Classic of Poetry, Book of Documents, Book of Rites, Book of Changes, Spring and Autumn Annals
3. Five Sacred Mountains in China: Mt. Huashan in Xian, Shaanxi, Mt. Taishan in Shandong, Mt. Hengshan in Hunan, Mt. Hengshan in Shanxi, Mt. Songshan in Henan
4. Five Emperors: Yellow Emperor (Emperor Huangdi), Emperor Zhuanxu, Emperor Diku, Yao, Shun

In the modern society, 5 is also widely used in some internet terms and is bestowed with new meanings by the youths. For example, in 7456, 5, with its Chinese pronunciation Wu which is similar to Wo, refers to I / me. 7456 means “It even makes me mad to death”. Similarly, 7758258 means “Kiss me now, and then love me”. 555555… with its pronunciation like whimpering, imitates a person’s sad cry.

Only 4 & 9. 4 is pronounced "shi" (死、death). 9 is "ku" (苦、suffering).

Exactly "5 is NOT" but not only "4" and "9".
Use google translate.
https://ja.wikipedia.org/wiki/%E5%BF%8C%E3%81%BF%E6%95%B0
http://wajikan.com/note/hukitusuuji/

Partially "5" is not good especially for worker.
http://d.hatena.ne.jp/keyword/%B8%DE%BD%BD%C6%FC

Just so that viewers who can't read Japanese know, Erena is not saying that five is an unlucky number in Japan. Even for workers, it's a great number (she points out that dates with five are busy dates at work).