Is watching free anime in Japan safe? I am talking about watching those shows in websites that provide them for free. I could of course afford to subscribe to paid websites such as Crunchyroll and the likes, but often I couldn't find the series I wanted to watch there. The free ones usually host a load of anime and drama titles.

Is watching free anime and Japanese drama in Japan considered safe?
In Japan or not, sites other than official sites that provides streaming for copyrighted/law protected contents are illegal and against the law. Thus, the answer to your question is no, it is not safe, it is against the law, it is a crime.

From IT (information tech) point of view, many such sites are actually crap sites that is full of ads, click baits, and links that will only redirect you around without showing the content if any. Such sites might also send adwares, malwares and the like to your computer. It might also be used to stole your identity. Thus it is also not safe from IT point of view.

If you are in Japan, however, you can watch the anime when it was broadcasted. This is perfectly legal. In fact, you can also record it if you want to watch later. AFAIK from anime that I watched, this is also legal. Some anime shows that recording a broadcasted show is okay. For example, in Futsuu no Joshikousei ga [Locodol] Yattemita., Yukari recorded a show that is being broadcasted. In Doraemon, Doraemon and Nobita would often record a show since it is being broadcasted at the same time as a baseball match that the father likes to watch.

Of course since it is limited to broad-casted anime only, you won't be able to watch older animes.