Here we have gathered together a few websites where you can listen to Japanese literature, folk stories and fairy tales and learn Japanese at the same time.

福娘童話集 Hukumusume Fairy Tale Collection (Introduction article) Today's Aesop's Fable

Here there are 366 days' worth of Aesop's Fables which you can listen to read aloud as you view the text, so since you can continue with a little every day it makes for useful Japanese listening practice.

Amongst the stories there are some read aloud in English. Press the Play button to hear the audio. Click on 'お話を表示 (Show story)' and the text will be displayed for you. You can even download the audio.

If you want to have the text displayed, you can choose between 'ひらがな (Hiragana)', '日本語・英語 (Japanese/English)' or 'English' displays. You can use the English display options to help you check the meanings as you read and listen to the text. '日本語・英語' gives you both English and Japanese displayed at the same time. 'English' is English only.

As well as there being 366 days' worth of tales, the length of the audio is also relatively short, which we think makes it easy to stick to a daily listening practice routine.

The message of the story is summed up and written at the end of each tale. This is a fun way for you to practice summarising the contents of the written piece having read the story, and to express the key points in Japanese.

デジタル絵本サイト Digital EHON (Picture Book) Site (Introduction article)

This site's pull is that you can read and listen to Japanese folklore tales in 12 different languages, with accompanying pictures.
There are 12 languages available on the menu, but the number of stories available in each differs. The Japanese language has the most stories available. There are some stories where you can listen and read the script in both Japanese and English.
In order to listen to the stories you need to download the audio files.
Learners who aren't yet fully confident in their Japanese skills can use the English script alongside the Japanese as they study, and there are pictures too to help make it more fun.

三重のむかしばなし Mie Fairy Tales

This website forms part of the contents of かんこうみえ.
Here you can watch some cheery animations, not of literary works, but folk tales passed down through the generations in Mie prefecture.
Text is displayed along with the animations.
If there are expressions you don't understand you can search for them in the text, then look up the meanings.

Otatsuhime

You cannot pause the animations. Also, since you cannot use online tools that let you attach ruby furigana to the text, this website is more suited to advanced learners.
The voices used are nice and clear, there are sound effects and pretty illustrations, so you can enjoy getting to know these Mie-based folk tales.
This would seem to be a good website for people who have an interest in Japanese fairy tales and want to put their Japanese ability to the test.

So, with this selection of websites you have a great deal of choice. Whether you want children's stories,  folk tales, animation, picture books, English and/or Japanese text display, downloadable or non-downloadable stories, each website offers something different.
Please try choosing the best options to suit your own needs!