From 23-24/06/2018 Telegram added this option, as part of the GDPR changes.
Now you can request to download all you data:
- Unfortunately it's not available yet (and not sure when) on the Telegram WebApp.
- Other platforms support for Data Export:
- Telegram Desktop - from 1.3.8 or higher
- Telegram for Android - from 4.8.10 or higher
- Telegram for iOS - from 4.8.3 or higher
You can message the @GDPRbot to request your data export.
Note: For security reasons, the data export is not immediate, so in case someone take control over one of your devices they won't be able to immediately download all your data.
Full notice from FAQ export:
Removing data on frequently messaged contacts and inline bots
To populate the ‘People’ box at the top of the Search section, Telegram calculates which of your chat partners you are likelier to message frequently. We calculate a similar rating for inline bots – thanks to that, the app can suggest you the bots you use frequently whenever you open the attachment menu (or start a new message with “@”).
We've implemented a way for you to remove this data from Telegram servers and stop Telegram from collecting it. It will become available in version 4.8.3 for iOS and version 4.8.10 for Android. Unfortunately, the update is still under review at the moment, but we expect it to become available in the nearest future.
Once you get the update, go to Settings > Privacy & Security and disable ‘Suggest Frequent Contacts’ to stop calculating ratings for contacts and inline bots, and delete all relevant data.
If you have other questions about data privacy, please use our @GDPRbot to contact Telegram's Data Protection Officer.
Demonstration from Telegram Desktop